Type:	Package
Title:	Computes Value at Risk and Expected Shortfall for over 100 Parametric Distributions
Version:	1.0.2
Date:	2023-04-20
Depends:	R (≥ 2.15.0)
Description:	Computes Value at risk and expected shortfall, two most popular measures of financial risk, for over one hundred parametric distributions, including all commonly known distributions. Also computed are the corresponding probability density function and cumulative distribution function. See Chan, Nadarajah and Afuecheta (2015) <doi:10.1080/03610918.2014.944658> for more details.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation:	no
BugReports:	https://github.com/lbelzile/VaRES/issues/
Packaged:	2023-04-21 16:03:39 UTC; lbelzile
Author:	Leo Belzile [cre], Saralees Nadarajah [aut], Stephen Chan [aut], Emmanuel Afuecheta [aut]
Maintainer:	Leo Belzile <belzilel@gmail.com>
Repository:	CRAN
Date/Publication:	2023-04-22 00:42:37 UTC

Computes value at risk and expected shortfall for over 100 parametric distributions

Description

Computes Value at risk and expected shortfall, two most popular measures of financial risk, for over one hundred parametric distributions, including all commonly known distributions. Also computed are the corresponding probability density function and cumulative distribution function.

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Birnbaum-Saunders distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Birnbaum-Saunders distribution due to Birnbaum and Saunders (1969a, 1969b) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {x^{1/2} + x^{-1/2}}{2 \gamma x} \phi \left( \frac {x^{1/2} - x^{-1/2}}{\gamma} \right), \\ &\displaystyle F (x) = \Phi \left( \frac {x^{1/2} - x^{-1/2}}{\gamma} \right), \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{4} \left\{ \gamma \Phi^{-1} (p) + \sqrt{4 + \gamma^2 \left[ \Phi^{-1} (p) \right]^2} \right\}^2, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{4 p} \int_0^p \left\{ \gamma \Phi^{-1} (v) + \sqrt{4 + \gamma^2 \left[ \Phi^{-1} (v) \right]^2} \right\}^2 dv \end{array}

for x > 0, 0 < p < 1, and \gamma > 0, the scale parameter.

Usage

dBS(x, gamma=1, log=FALSE)
pBS(x, gamma=1, log.p=FALSE, lower.tail=TRUE)
varBS(p, gamma=1, log.p=FALSE, lower.tail=TRUE)
esBS(p, gamma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dBS(x)
pBS(x)
varBS(x)
esBS(x)

Cauchy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Cauchy distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\pi} \frac {\sigma}{(x - \mu)^2 + \sigma^2}, \\ &\displaystyle F (x) = \frac {1}{2} + \frac {1}{\pi} \arctan \left( \frac {x - \mu}{\sigma} \right), \\ &\displaystyle {\rm VaR}_p (X) = \mu + \sigma \tan \left( \pi \left( p - \frac {1}{2} \right) \right), \\ &\displaystyle {\rm ES}_p (X) = \mu + \frac {\sigma}{p} \int_0^p \tan \left( \pi \left( v - \frac {1}{2} \right) \right) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter.

Usage

dCauchy(x, mu=0, sigma=1, log=FALSE)
pCauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varCauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esCauchy(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dCauchy(x)
pCauchy(x)
varCauchy(x)
esCauchy(x)

F distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the F distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{B \left( \frac {d_1}{2}, \frac {d_2}{2} \right)} \left( \frac {d_1}{d_2} \right)^{\frac {d_1}{2}} x^{\frac {d_1}{2} - 1} \left( 1 + \frac {d_1}{d_2} x \right)^{-\frac {d_1 + d_2}{2}}, \\ &\displaystyle F (x) = I_{\frac {d_1 x}{d_1 x + d_2}} \left( \frac {d_1}{2}, \frac {d_2}{2} \right), \\ &\displaystyle {\rm VaR}_p (X) = \frac {d_2}{d_1} \frac {I_p^{-1} \left( \frac {d_1}{2}, \frac {d_2}{2} \right)} {1 - I_p^{-1} \left( \frac {d_1}{2}, \frac {d_2}{2} \right)}, \\ &\displaystyle {\rm ES}_p (X) = \frac {d_2}{d_1 p} \int_0^p \frac {I_v^{-1} \left( \frac {d_1}{2}, \frac {d_2}{2} \right)} {1 - I_v^{-1} \left( \frac {d_1}{2}, \frac {d_2}{2} \right)} dv \end{array}

for x \geq K, 0 < p < 1, d_1 > 0, the first degree of freedom parameter, and d_2 > 0, the second degree of freedom parameter.

Usage

dF(x, d1=1, d2=1, log=FALSE)
pF(x, d1=1, d2=1, log.p=FALSE, lower.tail=TRUE)
varF(p, d1=1, d2=1, log.p=FALSE, lower.tail=TRUE)
esF(p, d1=1, d2=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dF(x)
pF(x)
varF(x)
esF(x)

Freimer distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Freimer distribution due to Freimer et al. (1988) given by

\begin{array}{ll} &\displaystyle {\rm VaR}_p (X) = \frac {1}{a} \left[ \frac {p^b - 1}{b} - \frac {(1 - p)^c - 1}{c} \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{a} \left( \frac {1}{c} - \frac {1}{b} \right) + \frac {p^b}{a b (b + 1)} + \frac {(1 - p)^{c + 1} - 1}{p a c (c + 1)} \end{array}

for 0 < p < 1, a > 0, the scale parameter, b > 0, the first shape parameter, and c > 0, the second shape parameter.

Usage

varFR(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
esFR(p, a=1, b=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
varFR(x)
esFR(x)

Gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the gamma distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b^a x^{a - 1} \exp (-b x)}{\Gamma (a)}, \\ &\displaystyle F (x) = \frac {\gamma (a, b x)}{\Gamma (a)}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} Q^{-1} (a, 1 - p), \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{b p} \int_0^p Q^{-1} (a, 1 - v) dv \end{array}

for x > 0, 0 < p < 1, b > 0, the scale parameter, and a > 0, the shape parameter, where \gamma (a, x) = \int_0^x t^{a - 1} \exp \left( -t \right) dt denotes the incomplete gamma function, Q (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt / \Gamma (a) denotes the regularized complementary incomplete gamma function, \Gamma (a) = \int_0^\infty t^{a - 1} \exp \left( -t \right) dt denotes the gamma function, and Q^{-1} (a, x) denotes the inverse of Q (a, x).

Usage

dGamma(x, a=1, b=1, log=FALSE)
pGamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varGamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esGamma(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dGamma(x)
pGamma(x)
varGamma(x)
esGamma(x)

Holla-Bhattacharya Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Holla-Bhattacharya Laplace distribution due to Holla and Bhattacharya (1968) given by

\begin{array}{ll} & f (x) = \left\{ \begin{array}{ll} \displaystyle a \phi \exp \left\{ \phi \left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle \left( 1 - a \right) \phi \exp \left\{ \phi \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,} \end{array} \right. \\ & F (x) = \left\{ \begin{array}{ll} \displaystyle a \exp \left( \phi x - \theta \phi \right), & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle 1 - (1 - a) \exp \left( \theta \phi - \phi x \right), & \mbox{if $x > \theta$,} \end{array} \right. \\ & {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta + \frac {1}{\phi} \log \left( \frac {p}{a} \right), & \mbox{if $p \leq a$,} \\ \\ \displaystyle \theta - \frac {1}{\phi} \log \left( \frac {1 - p}{1 - a} \right), & \mbox{if $p > a$,} \end{array} \right. \\ & {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \frac {1}{\phi} + \frac {1}{\phi} \log \frac {p}{a}, & \mbox{if $p \leq a$,} \\ \\ \displaystyle \frac {1}{p} \left[ \theta (1 + p - a) + \frac {p - 2a - (1 - a) \log a}{\phi} + \frac {1 - p}{\phi} \log \frac {1 - p}{1 - a} \right], & \mbox{if $p > a$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, 0 < a < 1, the first scale parameter, and \phi > 0, the second scale parameter.

Usage

dHBlaplace(x, a=0.5, theta=0, phi=1, log=FALSE)
pHBlaplace(x, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
varHBlaplace(p, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
esHBlaplace(p, a=0.5, theta=0, phi=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dHBlaplace(x)
pHBlaplace(x)
varHBlaplace(x)
esHBlaplace(x)

Hankin-Lee distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Hankin-Lee distribution due to Hankin and Lee (2006) given by

\begin{array}{ll} &\displaystyle {\rm VaR}_p (X) = \frac {c p^a}{(1 - p)^b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {c}{p} B_p (a + 1, 1 - b) \end{array}

for 0 < p < 1, c > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

varHL(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
esHL(p, a=1, b=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
varHL(x)
esHL(x)

Hosking logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Hosking logistic distribution due to Hosking (1989, 1990) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {(1 - k x)^{1 / k - 1}}{\left[ 1 + (1 - k x)^{1 / k} \right]^2}, \\ &\displaystyle F (x) = \frac {1}{1 + (1 - k x)^{1 / k}}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{k} \left[ 1 - \left( \frac {1 - p}{p} \right)^k \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{k} - \frac {1}{kp} B_p (1 - k, 1 + k) \end{array}

for x < 1/k if k > 0, x > 1/k if k < 0, -\infty < x < \infty if k = 0, and -\infty < k < \infty, the shape parameter.

Usage

dHlogis(x, k=1, log=FALSE)
pHlogis(x, k=1, log.p=FALSE, lower.tail=TRUE)
varHlogis(p, k=1, log.p=FALSE, lower.tail=TRUE)
esHlogis(p, k=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dHlogis(x)
pHlogis(x)
varHlogis(x)
esHlogis(x)

Libby-Novick beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Libby-Novick beta distribution due to Libby and Novick (1982) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\lambda^a x^{a - 1} (1 - x)^{b - 1}} {B (a, b) \left[ 1 - (1 - \lambda) x \right]^{a + b}}, \\ &\displaystyle F (x) = I_{\frac {\lambda x}{1 + (\lambda - 1) x}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \frac {I_p^{-1} (a, b)}{\lambda - (\lambda - 1) I_p^{-1} (a, b)}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \frac {I_v^{-1} (a, b)}{\lambda - (\lambda - 1) I_v^{-1} (a, b)} dv \end{array}

for 0 < x < 1, 0 < p < 1, \lambda > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dLNbeta(x, lambda=1, a=1, b=1, log=FALSE)
pLNbeta(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varLNbeta(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esLNbeta(p, lambda=1, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dLNbeta(x)
pLNbeta(x)
varLNbeta(x)
esLNbeta(x)

McDonald-Richards beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McDonald-Richards beta distribution due to McDonald and Richards (1987a, 1987b) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {x^{ar - 1} \left( bq^r - x^r \right)^{b - 1}} {\left( b q^r \right)^{a + b - 1} B (a, b)}, \\ &\displaystyle F (x) = I_{\frac {x^r}{b q^r}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = b^{1/r} q \left[ I_p^{-1} (a, b) \right]^{1/r}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b^{1/r} q}{p} \int_0^p \left[ I_v^{-1} (a, b) \right]^{1/r} dv \end{array}

for 0 \leq x \leq b^{1 / r} q, 0 < p < 1, q > 0, the scale parameter, a > 0, the first shape parameter, b > 0, the second shape parameter, and r > 0, the third shape parameter.

Usage

dMRbeta(x, a=1, b=1, r=1, q=1, log=FALSE)
pMRbeta(x, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)
varMRbeta(p, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)
esMRbeta(p, a=1, b=1, r=1, q=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dMRbeta(x)
pMRbeta(x)
varMRbeta(x)
esMRbeta(x)

McGill Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McGill Laplace distribution due to McGill (1962) given by

\begin{array}{ll} &\displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2 \psi} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle \frac {1}{2 \phi} \exp \left( \frac {\theta - x}{\phi} \right), & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle 1 - \frac {1}{2} \exp \left( \frac {\theta - x}{\phi} \right), & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta + \psi \log (2 p), & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta - \phi \log \left( 2 (1 - p) \right), & \mbox{if $p > 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \psi + \theta \log (2 p) - \theta p, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta + \phi + \frac {\psi - \phi - 2 \theta}{2 p} + \frac {\phi}{p} \log 2 - \phi \log 2 \\ \displaystyle \quad +\frac {\phi}{p} \log (1 - p) - \phi \log (1 - p), & \mbox{if $p > 1/2$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, \phi > 0, the first scale parameter, and \psi > 0, the second scale parameter.

Usage

dMlaplace(x, theta=0, phi=1, psi=1, log=FALSE)
pMlaplace(x, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)
varMlaplace(p, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)
esMlaplace(p, theta=0, phi=1, psi=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dMlaplace(x)
pMlaplace(x)
varMlaplace(x)
esMlaplace(x)

Poiraud-Casanova-Thomas-Agnan Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Poiraud-Casanova-Thomas-Agnan Laplace distribution due to Poiraud-Casanova and Thomas-Agnan (2000) given by

\begin{array}{ll} &\displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle a \left( 1 - a \right) \exp \left\{ \left( 1 - a \right) \left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle a \left( 1 - a \right) \exp \left\{ a \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle a \exp \left\{ \left( 1 - a \right) \left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle 1 - \left( 1 - a \right) \exp \left\{ a \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta + \frac {1}{1 - a} \log \left( \frac {p}{a} \right), & \mbox{if $p \leq a$,} \\ \\ \displaystyle \theta - \frac {1}{a} \log \left( \frac {1 - p}{1 - a} \right), & \mbox{if $p > a$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \frac {\log a}{1 - a} + \frac {\log p - 1}{(1 - a) p}, & \mbox{if $p \leq a$,} \\ \\ \displaystyle \theta - \frac {1}{a} + \frac {1}{p} - \frac {a}{(1 - a) p} + \frac {1 - p}{a p} \log \left( \frac {1 - p}{1 - a} \right), & \mbox{if $p > a$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, and a > 0, the scale parameter.

Usage

dPCTAlaplace(x, a=0.5, theta=0, log=FALSE)
pPCTAlaplace(x, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)
varPCTAlaplace(p, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)
esPCTAlaplace(p, a=0.5, theta=0)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dPCTAlaplace(x)
pPCTAlaplace(x)
varPCTAlaplace(x)
esPCTAlaplace(x)

Ramberg-Schmeiser distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Ramber-Schmeiser distribution due to Ramberg and Schmeiser (1974) given by

\begin{array}{ll} &\displaystyle {\rm VaR}_p (X) = \frac {p^b - (1 - p)^c}{d}, \\ &\displaystyle {\rm ES}_p (X) = \frac {p^{b}}{d (b + 1)} + \frac {(1 - p)^{c + 1} - 1}{p d (c + 1)} \end{array}

for 0 < p < 1, b > 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, the scale parameter.

Usage

varRS(p, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
esRS(p, b=1, c=1, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
varRS(x)
esRS(x)

Student's t distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Student's t distribution due to Gosset (1908) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\Gamma \left( \frac {n + 1}{2} \right)}{\sqrt{n \pi} \Gamma \left( \frac {n}{2} \right)} \left( 1 + \frac {x^2}{n} \right)^{-\frac {n + 1}{2}}, \\ &\displaystyle F (x) = \frac {1 + {\rm sign} (x)}{2} - \frac {{\rm sign} (x)}{2} I_{\frac {n}{x^2 + n}} \left( \frac {n}{2}, \frac {1}{2} \right), \\ &\displaystyle {\rm VaR}_p (X) = \sqrt{n} {\rm sign} \left( p - \frac {1}{2} \right) \sqrt{\frac {1}{I_a^{-1} \left( \frac {n}{2}, \frac {1}{2} \right)} - 1}, \\ &\displaystyle \quad \mbox{ where $a = 2p$ if $p < 1/2$, $a = 2(1 - p)$ if $p \geq 1/2$,} \\ &\displaystyle {\rm ES}_p (X) = \frac {\sqrt{n}}{p} \int_0^p {\rm sign} \left( v - \frac {1}{2} \right) \sqrt{\frac {1}{I_a^{-1} \left( \frac {n}{2}, \frac {1}{2} \right)} - 1} dv, \\ &\displaystyle \quad \mbox{ where $a = 2v$ if $v < 1/2$, $a = 2(1 - v)$ if $v \geq 1/2$} \end{array}

for -\infty < x < \infty, 0 < p < 1, and n > 0, the degree of freedom parameter.

Usage

dT(x, n=1, log=FALSE)
pT(x, n=1, log.p=FALSE, lower.tail=TRUE)
varT(p, n=1, log.p=FALSE, lower.tail=TRUE)
esT(p, n=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dT(x)
pT(x)
varT(x)
esT(x)

Tukey-Lambda distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Tukey-Lambda distribution due to Tukey (1962) given by

\begin{array}{ll} &\displaystyle {\rm VaR}_p (X) = \frac {p^\lambda - (1 - p)^\lambda}{\lambda}, \\ &\displaystyle {\rm ES}_p (X) = \frac {p^{\lambda + 1} + (1 - p)^{\lambda + 1} - 1}{p \lambda (\lambda + 1)} \end{array}

for 0 < p < 1, and \lambda > 0, the shape parameter.

Usage

varTL(p, lambda=1, log.p=FALSE, lower.tail=TRUE)
esTL(p, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
varTL(x)
esTL(x)

Topp-Leone distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Topp-Leone distribution due to Topp and Leone (1955) given by

\begin{array}{ll} &\displaystyle f(x) = 2 b (x (2 - x))^{b - 1} (1 - x), \\ &\displaystyle F(x) = (x (2 - x))^b, \\ &\displaystyle {\rm VaR}_p (X) = 1 - \sqrt{1 - p^{1 / b}}, \\ &\displaystyle {\rm ES}_p (X) = 1 - \frac {b}{p} B_{p^{1 / b}} \left( b, \frac {3}{2} \right) \end{array}

for x > 0, 0 < p < 1, and b > 0, the shape parameter.

Usage

dTL2(x, b=1, log=FALSE)
pTL2(x, b=1, log.p=FALSE, lower.tail=TRUE)
varTL2(p, b=1, log.p=FALSE, lower.tail=TRUE)
esTL2(p, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dTL2(x)
pTL2(x)
varTL2(x)
esTL2(x)

Asymmetric exponential power distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric exponential power distribution due to Zhu and Zinde-Walsh (2009) given by

\begin{array}{ll} &\displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {\alpha}{\alpha^{*}} K \left( q_1 \right) \exp \left[ -\frac {1}{q_1} \left | \frac {x}{2 \alpha^{*}} \right |^{q_1} \right], & \mbox{if $x \leq 0$,} \\ \\ \displaystyle \frac {1 - \alpha}{1 - \alpha^{*}} K \left( q_2 \right) \exp \left[ -\frac {1}{q_2} \left | \frac {x}{2 - 2 \alpha^{*}} \right |^{q_2} \right], & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \alpha Q \left( \frac {1}{q_1} \left( \frac {\mid x \mid}{2 \alpha^{*}} \right)^{q_1}, \frac {1}{q_1} \right), & \mbox{if $x \leq 0$,} \\ \\ \displaystyle 1 - (1 - \alpha) Q \left( \frac {1}{q_2} \left( \frac {\mid x \mid}{2 - 2 \alpha^{*}} \right)^{q_2}, \frac {1}{q_2} \right), & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle -2 \alpha^{*} \left[ q_1 Q^{-1} \left( \frac {p}{\alpha}, \frac {1}{q_1} \right) \right]^{\frac {1}{q_1}}, & \mbox{if $p \leq \alpha$,} \\ \\ \displaystyle 2 \left(1 - \alpha^{*}\right) \left[ q_2 Q^{-1} \left( \frac {1 - p}{1 - \alpha}, \frac {1}{q_2} \right) \right]^{\frac {1}{q_2}}, & \mbox{if $p > \alpha$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle -\frac {2 \alpha^{*}}{p} \int_0^p \left[ q_1 Q^{-1} \left( \frac {v}{\alpha}, \frac {1}{q_1} \right) \right]^{\frac {1}{q_1}} dv, & \mbox{if $p \leq \alpha$,} \\ \\ \displaystyle -\frac {2 \alpha^{*}}{p} \int_0^\alpha \left[ q_1 Q^{-1} \left( \frac {v}{\alpha}, \frac {1}{q_1} \right) \right]^{\frac {1}{q_1}} dv & \ \\ \quad \displaystyle +\frac {2 \left(1 - \alpha^{*}\right)}{p} \int_\alpha^p \left[ q_2 Q^{-1} \left( \frac {1 - v}{1 - \alpha}, \frac {1}{q_2} \right) \right]^{\frac {1}{q_2}} dv, & \mbox{if $p > \alpha$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, 0 < \alpha < 1, the scale parameter, q_1 > 0, the first shape parameter, and q_2 > 0, the second shape parameter, where \alpha^{*} = \alpha K \left( q_1 \right) / \left\{ \alpha K \left( q_1 \right) + (1 - \alpha) K \left( q_2 \right) \right\}, K (q) = \frac {1}{2 q^{1/q} \Gamma (1 + 1/q)}, Q (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt / \Gamma (a) denotes the regularized complementary incomplete gamma function, \Gamma (a) = \int_0^\infty t^{a - 1} \exp \left( -t \right) dt denotes the gamma function, and Q^{-1} (a, x) denotes the inverse of Q (a, x).

Usage

daep(x, q1=1, q2=1, alpha=0.5, log=FALSE)
paep(x, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
varaep(p, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
esaep(p, q1=1, q2=1, alpha=0.5)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
daep(x)
paep(x)
varaep(x)
esaep(x)

Arcsine distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the arcsine distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\pi \sqrt{(x - a) (b - x)}}, \\ &\displaystyle F (x) = \frac {2}{\pi} \arcsin \left( \sqrt{\frac {x - a}{b - a}} \right), \\ &\displaystyle {\rm VaR}_p (X) = a + (b - a) \sin^2 \left( \frac {\pi p}{2} \right), \\ &\displaystyle {\rm ES}_p (X) = a + \frac {b - a}{p} \int_0^p \sin^2 \left( \frac {\pi v}{2} \right) dv \end{array}

for a \leq x \leq b, 0 < p < 1, -\infty < a < \infty, the first location parameter, and -\infty < a < b < \infty, the second location parameter.

Usage

darcsine(x, a=0, b=1, log=FALSE)
parcsine(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)
vararcsine(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)
esarcsine(p, a=0, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
darcsine(x)
parcsine(x)
vararcsine(x)
esarcsine(x)

Generalized asymmetric Student's t distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized asymmetric Student's t distribution due to Zhu and Galbraith (2010) given by

\begin{array}{ll} &\displaystyle \displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {\alpha}{\alpha^{*}} K \left( \nu_1 \right) \left[ 1 + \frac {1}{\nu_1} \left( \frac {x}{2 \alpha^{*}} \right)^2 \right]^{-\frac {\nu_1 + 1}{2}}, & \mbox{if $x \leq 0$,} \\ \\ \displaystyle \frac {1 - \alpha}{1 - \alpha^{*}} K \left( \nu_2 \right) \left[ 1 + \frac {1}{\nu_2} \left( \frac {x}{2 \left( 1 - \alpha^{*} \right)} \right)^2 \right]^{-\frac {\nu_2 + 1}{2}}, & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle \displaystyle F (x) = 2 \alpha F_{\nu_1} \left( \frac {\min (x, 0)}{2 \alpha^{*}} \right) -1 + \alpha + 2 (1 - \alpha) F_{\nu_2} \left( \frac {\max (x, 0)}{2 - 2 \alpha^{*}} \right), \\ &\displaystyle \displaystyle {\rm VaR}_p (X) = 2 \alpha^{*} F_{\nu_1}^{-1} \left( \frac {\min (p, \alpha)}{2 \alpha} \right) + 2 \left( 1 - \alpha^{*} \right) F_{\nu_2}^{-1} \left( \frac {\max (p, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right), \\ &\displaystyle \displaystyle {\rm ES}_p (X) = \frac {2 \alpha^{*}}{p} \int_0^p F_{\nu_1}^{-1} \left( \frac {\min (v, \alpha)}{2 \alpha} \right) dv + \frac {2 \left( 1 - \alpha^{*} \right)}{p} \int_0^p F_{\nu_2}^{-1} \left( \frac {\max (v, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, 0 < \alpha < 1, the scale parameter, \nu_1 > 0, the first degree of freedom parameter, and \nu_2 > 0, the second degree of freedom parameter, where \alpha^{*} = \alpha K \left( \nu_1 \right) / \left\{ \alpha K \left( \nu_1 \right) + (1 - \alpha) K \left( \nu_2 \right) \right\}, K (\nu) = \Gamma \left( (\nu + 1)/2 \right) / \left[ \sqrt{\pi \nu} \Gamma (\nu/2) \right], F_\nu(\cdot) denotes the cdf of a Student's t random variable with \nu degrees of freedom, and F_\nu^{-1} (\cdot) denotes the inverse of F_\nu(\cdot).

Usage

dast(x, nu1=1, nu2=1, alpha=0.5, log=FALSE)
past(x, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
varast(p, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
esast(p, nu1=1, nu2=1, alpha=0.5)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dast(x)
past(x)
varast(x)
esast(x)

Asymmetric Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric Laplace distribution due to Kotz et al. (2001) given by

\begin{array}{ll} &\displaystyle \displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac {\kappa \sqrt{2}}{\tau \left( 1 + \kappa^2 \right)} \exp \left( -\frac {\kappa \sqrt{2}}{\tau} \left | x - \theta \right | \right), & \mbox{if $x \geq \theta$,} \\ \\ \displaystyle \frac {\kappa \sqrt{2}}{\tau \left( 1 + \kappa^2 \right)} \exp \left( -\frac {\sqrt{2}}{\kappa \tau} \left | x - \theta \right | \right), & \mbox{if $x < \theta$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle 1 - \frac {1}{1 + \kappa^2} \exp \left( \frac {\kappa \sqrt{2} (\theta - x)}{\tau} \right), & \mbox{if $x \geq \theta$,} \\ \\ \displaystyle \frac {\kappa^2}{1 + \kappa^2} \exp \left( \frac {\sqrt{2} (x - \theta)}{\kappa \tau} \right), & \mbox{if $x < \theta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \frac {\tau}{\sqrt{2} \kappa} \log \left[ (1 - p) \left( 1 + \kappa^2 \right) \right], & \mbox{if $p \geq \frac {\kappa^2}{1 + \kappa^2}$,} \\ \\ \displaystyle \theta + \frac {\kappa \tau}{\sqrt{2}} \log \left[ p \left( 1 + \kappa^{-2} \right) \right], & \mbox{if $p < \frac {\kappa^2}{1 + \kappa^2}$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \frac {\theta}{p} + \theta - \frac {\tau}{\sqrt{2} \kappa} \log \left( 1 + \kappa^2 \right) + \frac {\sqrt{2} \tau \left( 1 + 2 \kappa^2 \right)}{2 \kappa \left( 1 + \kappa^2 \right) p} \log \left( 1 + \kappa^2 \right) \\ \displaystyle \quad -\frac {\sqrt{2} \tau \kappa \log \kappa}{\left( 1 + \kappa^2 \right) p} - \frac {\theta \kappa^2}{\left( 1 + \kappa^2 \right) p} + \frac {\tau \left( 1 - \kappa^4 \right)}{\sqrt{2} \kappa \left( 1 + \kappa^2 \right) p} \\ \displaystyle \quad -\frac {\tau (1 - p)}{\sqrt{2} \kappa p} + \frac {\tau (1 - p)}{\sqrt{2} \kappa p} \log (1 - p), & \mbox{if $p \geq \frac {\kappa^2}{1 + \kappa^2}$,} \\ \\ \displaystyle \theta + \frac {\kappa \tau}{\sqrt{2}} \log \left( 1 + \kappa^{-2} \right) + \frac {\kappa \tau}{\sqrt{2}} (\log p - 1), & \mbox{if $p < \frac {\kappa^2}{1 + \kappa^2}$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, \kappa > 0, the first scale parameter, and \tau > 0, the second scale parameter.

Usage

dasylaplace(x, tau=1, kappa=1, theta=0, log=FALSE)
pasylaplace(x, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)
varasylaplace(p, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)
esasylaplace(p, tau=1, kappa=1, theta=0)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dasylaplace(x)
pasylaplace(x)
varasylaplace(x)
esasylaplace(x)

Asymmetric power distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric power distribution due to Komunjer (2007) given by

\begin{array}{ll} &\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac {\displaystyle \delta^{1 / \lambda}}{\displaystyle \Gamma (1 + 1 / \lambda)} \exp \left[ -\frac {\delta}{a^\lambda} |x|^\lambda \right], & \mbox{if $x \leq 0$}, \\ \\ \displaystyle \frac {\displaystyle \delta^{1 / \lambda}}{\displaystyle \Gamma (1 + 1 / \lambda)} \exp \left[ -\frac {\delta}{(1 - a)^\lambda} |x|^\lambda \right], & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle a - a {\cal I} \left( \frac {\delta}{a^\lambda} \sqrt{\lambda} |x|^\lambda, 1 / \lambda \right), & \mbox{if $x \leq 0$,} \\ \\ \displaystyle a - (1 - a) {\cal I} \left( \frac {\delta}{(1 - a)^\lambda} \sqrt{\lambda} |x|^\lambda, 1 / \lambda \right), & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle -\left[ \frac {a^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda} \left[ {\cal I}^{-1} \left( 1 - \frac {p}{a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda}, & \mbox{if $p \leq a$,} \\ \\ \displaystyle -\left[ \frac {(1 - a)^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda} \left[ {\cal I}^{-1} \left( 1 - \frac {1 - p}{1 - a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda}, & \mbox{if $p > a$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle -\frac {1}{p} \left[ \frac {a^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda} \int_0^p \left[ {\cal I}^{-1} \left( 1 - \frac {v}{a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda} dv, & \mbox{if $p \leq a$,} \\ \\ \displaystyle -\frac {1}{p} \left[ \frac {a^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda} \int_0^a \left[ {\cal I}^{-1} \left( 1 - \frac {v}{a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda} dv \\ \quad \displaystyle -\frac {1}{p} \left[ \frac {(1 - a)^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda} \int_a^p \left[ {\cal I}^{-1} \left( 1 - \frac {1 - v}{1 - a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda} dv, & \mbox{if $p > a$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, 0 < a < 1, the first scale parameter, \delta > 0, the second scale parameter, and \lambda > 0, the shape parameter, where {\cal I} (x, \gamma) = \frac {1}{\Gamma (\gamma)} \int_0^{x \sqrt{\gamma}} t^{\gamma - 1} \exp (-t) dt.

Usage

dasypower(x, a=0.5, lambda=1, delta=1, log=FALSE)
pasypower(x, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)
varasypower(p, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)
esasypower(p, a=0.5, lambda=1, delta=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dasypower(x)
pasypower(x)
varasypower(x)
esasypower(x)

Beard distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Beard distribution due to Beard (1959) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {\displaystyle a \exp (b x) \left[ 1 + a \rho \right]^{\rho^{-1/b}}} {\displaystyle \left[ 1 + a \rho \exp (b x) \right]^{1 + \rho^{-1/b}}}, \\ &\displaystyle F (x) = 1 - \frac {\displaystyle \left[ 1 + a \rho \right]^{\rho^{-1/b}}} {\displaystyle \left[ 1 + a \rho \exp (b x) \right]^{\rho^{-1/b}}}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} \log \left[ \frac {1 + a \rho}{a \rho (1 - p)^{\rho^{1 / b}}} - \frac {1}{a \rho} \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p b} \int_0^p \log \left[ -\frac {1}{a \rho} + \frac {1 + a \rho}{a \rho (1 - v)^{\rho^{1 / b}}} \right] dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first scale parameter, b > 0, the second scale parameter, and \rho > 0, the shape parameter.

Usage

dbeard(x, a=1, b=1, rho=1, log=FALSE)
pbeard(x, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)
varbeard(p, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)
esbeard(p, a=1, b=1, rho=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbeard(x)
pbeard(x)
varbeard(x)
esbeard(x)

Beta Burr distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr distribution due to Parana\'iba et al. (2011) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b a^{bd}}{B (c, d)x^{bd + 1}} \left[ 1 + \left( x / a \right)^{-b} \right]^{-c - d}, \\ &\displaystyle F (x) = I_{\frac {1}{1 + \left( x / a \right)^{-b}}} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = a \left[ I_p^{-1} (c, d) \right]^{1 / b} \left[ 1 - I_p^{-1} (c, d) \right]^{-1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {a}{p} \int_0^p \left[ I_v^{-1} (c, d) \right]^{1 / b} \left[ 1 - I_v^{-1} (c, d) \right]^{-1 / b} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the scale parameter, b > 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter, where I_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt / B (a, b) denotes the incomplete beta function ratio, B (a, b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt denotes the beta function, and I_x^{-1} (a, b) denotes the inverse function of I_x (a, b).

Usage

dbetaburr(x, a=1, b=1, c=1, d=1, log=FALSE)
pbetaburr(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
varbetaburr(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
esbetaburr(p, a=1, b=1, c=1, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetaburr(x)
pbetaburr(x)
varbetaburr(x)
esbetaburr(x)

Beta Burr XII distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr XII distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {k c x^{c - 1}}{B (a, b)} \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^{a - 1} \left( 1 + x^c \right)^{-b k - 1}, \\ &\displaystyle F (x) = I_{1 - \left( 1 + x^c \right)^{-k}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \left[ 1 - I_p^{-1} (a, b) \right]^{-1 / k} - 1 \right\}^{1/c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left\{ \left[ 1 - I_v^{-1} (a, b) \right]^{-1 / k} - 1 \right\}^{1/c} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, c > 0, the third shape parameter, and k > 0, the fourth shape parameter.

Usage

dbetaburr7(x, a=1, b=1, c=1, k=1, log=FALSE)
pbetaburr7(x, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE)
varbetaburr7(p, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE)
esbetaburr7(p, a=1, b=1, c=1, k=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetaburr7(x)
pbetaburr7(x)
varbetaburr7(x)
esbetaburr7(x)

Beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {x^{a - 1} (1 - x)^{b - 1}}{B (a, b)}, \\ &\displaystyle F (x) = I_x (a, b), \\ &\displaystyle {\rm VaR}_p (X) = I_p^{-1} (a, b), \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p I_v^{-1} (a, b) dv \end{array}

for 0 < x < 1, 0 < p < 1, a > 0, the first parameter, and b > 0, the second shape parameter.

Usage

dbetadist(x, a=1, b=1, log=FALSE)
pbetadist(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varbetadist(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esbetadist(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetadist(x)
pbetadist(x)
varbetadist(x)
esbetadist(x)

Beta exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta exponential distribution due to Nadarajah and Kotz (2006) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\lambda \exp (-b \lambda x)}{B (a, b)} \left[ 1 - \exp (-\lambda x) \right]^{a - 1}, \\ &\displaystyle F (x) = I_{1 - \exp (-\lambda x)} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{\lambda} \log \left[ 1 - I_p^{-1} (a, b) \right], \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{p \lambda} \int_0^p \log \left[ 1 - I_v^{-1} (a, b) \right] dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, and \lambda > 0, the scale parameter, where I_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt / B (a, b) denotes the incomplete beta function ratio, B (a, b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt denotes the beta function, and I_x^{-1} (a, b) denotes the inverse function of I_x (a, b).

Usage

dbetaexp(x, lambda=1, a=1, b=1, log=FALSE)
pbetaexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varbetaexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esbetaexp(p, lambda=1, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetaexp(x)
pbetaexp(x)
varbetaexp(x)
esbetaexp(x)

Beta Frechet distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Fr\'echet distribution due to Barreto-Souza et al. (2011) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\alpha \sigma^\alpha}{x^{\alpha + 1} B (a, b)} \exp \left\{ -a \left( \frac {\sigma}{x} \right)^{\alpha} \right\} \left[ 1 - \exp \left\{ -\left( \frac {\sigma}{x} \right)^{\alpha} \right\} \right]^{b - 1}, \\ &\displaystyle F (x) = I_{\exp \left\{ -\left( \frac {\sigma}{x} \right)^{\alpha} \right\}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \sigma \left[ -\log I_p^{-1} (a, b) \right]^{-1 / \alpha}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left[ -\log I_v^{-1} (a, b) \right]^{-1 / \alpha} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, \sigma > 0, the scale parameter, b > 0, the second shape parameter, and \alpha > 0, the third shape parameter.

Usage

dbetafrechet(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)
pbetafrechet(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varbetafrechet(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esbetafrechet(p, a=1, b=1, alpha=1, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetafrechet(x)
pbetafrechet(x)
varbetafrechet(x)
esbetafrechet(x)

Beta Gompertz distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gompertz distribution due to Cordeiro et al. (2012b) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {b \eta \exp (bx)}{B (c, d)} \exp \left( d \eta \right) \exp \left[ -d \eta \exp (bx) \right] \left\{ 1 - \exp \left[ \eta - \eta \exp (bx) \right] \right\}^{c - 1}, \\ &\displaystyle F(x) = I_{1 - \exp \left[ \eta - \eta \exp (bx) \right]} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} \log \left\{ 1 - \frac {1}{\eta} \log \left[ 1 - I_p^{-1} (c, d) \right] \right\}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p b} \int_0^p \log \left\{ 1 - \frac {1}{\eta} \log \left[ 1 - I_v^{-1} (c, d) \right] \right\} dv \end{array}

for x > 0, 0 < p < 1, b > 0, the first scale parameter, \eta > 0, the second scale parameter, c > 0, the first shape parameter, and d > 0, the second shape parameter.

Usage

dbetagompertz(x, b=1, c=1, d=1, eta=1, log=FALSE)
pbetagompertz(x, b=1, c=1, d=1, eta=1, log.p=FALSE, lower.tail=TRUE)
varbetagompertz(p, b=1, c=1, d=1, eta=1, log.p=FALSE, lower.tail=TRUE)
esbetagompertz(p, b=1, c=1, d=1, eta=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetagompertz(x)
pbetagompertz(x)
varbetagompertz(x)
esbetagompertz(x)

Beta Gumbel distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel distribution due to Nadarajah and Kotz (2004) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma B (a, b)} \exp \left( \frac {\mu - x}{\sigma} \right) \exp \left[ -a \exp \frac {\mu - x}{\sigma} \right] \left\{ 1 - \exp \left[ -\exp \frac {\mu - x}{\sigma} \right] \right\}^{b - 1}, \\ &\displaystyle F (x) = I_{\exp \left[ -\exp \frac {\mu - x}{\sigma} \right]} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \mu - \sigma \log \left[ -\log I_p^{-1} (a, b) \right], \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{p} \int_0^p \log \left[ -\log I_v^{-1} (a, b) \right] dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dbetagumbel(x, a=1, b=1, mu=0, sigma=1, log=FALSE)
pbetagumbel(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varbetagumbel(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esbetagumbel(p, a=1, b=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetagumbel(x)
pbetagumbel(x)
varbetagumbel(x)
esbetagumbel(x)

Beta Gumbel 2 distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel II distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b x^{-a - 1}}{B (c, d)} \exp \left( -b d x^{-a} \right) \left[ 1 - \exp \left( -b x^{-a} \right) \right]^{c - 1}, \\ &\displaystyle F (x) = I_{1 - \exp \left( -b x^{-a} \right)} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = b^{1 / a} \left\{ -\log \left[ 1 - I_p^{-1} (c, d) \right] \right\}^{-1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b^{1 / a}}{p} \int_0^p \left\{ -\log \left[ 1 - I_v^{-1} (c, d) \right] \right\}^{-1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter.

Usage

dbetagumbel2(x, a=1, b=1, c=1, d=1, log=FALSE)
pbetagumbel2(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
varbetagumbel2(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
esbetagumbel2(p, a=1, b=1, c=1, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetagumbel2(x)
pbetagumbel2(x)
varbetagumbel2(x)
esbetagumbel2(x, a = 2)

Beta lognormal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta lognormal distribution due to Castellares et al. (2013) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma x B (a, b)} \phi \left( \frac {\log x - \mu}{\sigma} \right) \Phi^{a - 1} \left( \frac {\log x - \mu}{\sigma} \right) \Phi^{b - 1} \left( \frac {\mu - \log x}{\sigma} \right), \\ &\displaystyle F (x) = I_{\Phi \left( \frac {\log x - \mu}{\sigma} \right)} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \exp \left[ \mu + \sigma \Phi^{-1} \left( I_p^{-1} (a, b) \right) \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {\exp (\mu)}{p} \int_0^p \exp \left[ \sigma \Phi^{-1} \left( I_v^{-1} (a, b) \right) \right] dv \end{array}

for x > 0, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter, where \phi (\cdot) denotes the pdf of a standard normal random variable, and \Phi (\cdot) denotes the cdf of a standard normal random variable.

Usage

dbetalognorm(x, a=1, b=1, mu=0, sigma=1, log=FALSE)
pbetalognorm(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varbetalognorm(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esbetalognorm(p, a=1, b=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetalognorm(x)
pbetalognorm(x)
varbetalognorm(x)
esbetalognorm(x)

Beta Lomax distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Lomax distribution due to Lemonte and Cordeiro (2013) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\alpha}{\lambda B (a, b)} \left( 1 + \frac {x}{\lambda} \right)^{-b \alpha - 1} \left[ 1 - \left( 1 + \frac {x}{\lambda} \right)^{-\alpha} \right]^{a - 1}, \\ &\displaystyle F (x) = I_{1 - \left( 1 + \frac {x}{\lambda} \right)^{-\alpha}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \lambda \left[ 1 - I_p^{-1} (a, b) \right]^{-1 / \alpha} - \lambda, \\ &\displaystyle {\rm ES}_p (X) = \frac {\lambda}{p} \int_0^p \left[ 1 - I_v^{-1} (a, b) \right]^{-1 / \alpha} dv - \lambda \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, \alpha > 0, the third shape parameter, and \lambda > 0, the scale parameter.

Usage

dbetalomax(x, a=1, b=1, alpha=1, lambda=1, log=FALSE)
pbetalomax(x, a=1, b=1, alpha=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varbetalomax(p, a=1, b=1, alpha=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
esbetalomax(p, a=1, b=1, alpha=1, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetalomax(x)
pbetalomax(x)
varbetalomax(x)
esbetalomax(x)

Beta normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta normal distribution due to Eugene et al. (2002) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma B (a, b)} \phi \left( \frac {x - \mu}{\sigma} \right) \Phi^{a - 1} \left( \frac {x - \mu}{\sigma} \right) \Phi^{b - 1} \left( \frac {\mu - x}{\sigma} \right), \\ &\displaystyle F (x) = I_{\Phi \left( \frac {x - \mu}{\sigma} \right)} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \mu + \sigma \Phi^{-1} \left( I_p^{-1} (a, b) \right), \\ &\displaystyle {\rm ES}_p (X) = \mu + \frac {\sigma}{p} \int_0^p \Phi^{-1} \left( I_v^{-1} (a, b) \right) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dbetanorm(x, mu=0, sigma=1, a=1, b=1, log=FALSE)
pbetanorm(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varbetanorm(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esbetanorm(p, mu=0, sigma=1, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetanorm(x)
pbetanorm(x)
varbetanorm(x)
esbetanorm(x)

Beta Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Pareto distribution due to Akinsete et al. (2008) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a K^{ad} x^{-ad - 1}}{B (c, d)} \left[ 1 - \left( \frac {K}{x} \right)^a \right]^{c - 1}, \\ &\displaystyle F (x) = I_{1 - \left( \frac {K}{x} \right)^a} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = K \left[ 1 - I_p^{-1} (c, d) \right]^{-1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {K}{p} \int_0^p \left[ 1 - I_v^{-1} (c, d) \right]^{-1 / a} dv \end{array}

for x \geq K, 0 < p < 1, K > 0, the scale parameter, a > 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter.

Usage

dbetapareto(x, K=1, a=1, c=1, d=1, log=FALSE)
pbetapareto(x, K=1, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
varbetapareto(p, K=1, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
esbetapareto(p, K=1, a=1, c=1, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetapareto(x)
pbetapareto(x)
varbetapareto(x)
esbetapareto(x)

Beta Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Weibull distribution due to Cordeiro et al. (2012b) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {\alpha x^{\alpha - 1}}{\sigma^\alpha B (a, b)} \exp \left\{ -b \left( \frac {x}{\sigma} \right)^{\alpha} \right\} \left[ 1 - \exp \left\{ -\left( \frac {x}{\sigma} \right)^{\alpha} \right\} \right]^{a - 1}, \\ &\displaystyle F(x) = I_{1 - \exp \left\{ -\left( \frac {x}{\sigma} \right)^{\alpha} \right\}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \sigma \left\{ -\log \left[ 1 - I_p^{-1} (a, b) \right] \right\}^{1 / \alpha}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left\{ -\log \left[ 1 - I_v^{-1} (a, b) \right] \right\}^{1 / \alpha} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, \alpha > 0, the third shape parameter, and \sigma > 0, the scale parameter.

Usage

dbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)
pbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varbetaweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esbetaweibull(p, a=1, b=1, alpha=1, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetaweibull(x)
pbetaweibull(x)
varbetaweibull(x)
esbetaweibull(x)

Burr distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Burr distribution due to Burr (1942) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b a^b}{x^{b + 1}} \left[ 1 + \left( x / a \right)^{-b} \right]^{-2}, \\ &\displaystyle F (x) = \frac {1}{1 + \left( x / a \right)^{-b}}, \\ &\displaystyle {\rm VaR}_p (X) = a p^{1 / b} (1 - p)^{-1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {a}{p} B_p \left( 1 / b + 1, 1 - 1 / b \right) \end{array}

for x > 0, 0 < p < 1, a > 0, the scale parameter, and b > 0, the shape parameter, where B_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt denotes the incomplete beta function.

Usage

dburr(x, a=1, b=1, log=FALSE)
pburr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varburr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esburr(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dburr(x)
pburr(x)
varburr(x)
esburr(x)

Burr XII distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Burr XII distribution due to Burr (1942) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {k c x^{c - 1}}{\left( 1 + x^c \right)^{k + 1}}, \\ &\displaystyle F (x) = 1 - \left( 1 + x^c \right)^{-k}, \\ &\displaystyle {\rm VaR}_p (X) = \left[ (1 - p)^{-1 / k} - 1 \right]^{1/c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ (1 - v)^{-1 / k} - 1 \right]^{1/c} dv \end{array}

for x > 0, 0 < p < 1, c > 0, the first shape parameter, and k > 0, the second shape parameter.

Usage

dburr7(x, k=1, c=1, log=FALSE)
pburr7(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
varburr7(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
esburr7(p, k=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dburr7(x)
pburr7(x)
varburr7(x)
esburr7(x)

Chen distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Chen distribution due to Chen (2000) given by

\begin{array}{ll} &\displaystyle f(x) = \lambda b x^{b - 1} \exp \left( x^b \right) \exp \left[ \lambda - \lambda \exp \left( x^b \right) \right], \\ &\displaystyle F (x) = 1 - \exp \left[ \lambda - \lambda \exp \left( x^b \right) \right], \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \log \left[ 1 - \frac {\log (1 - p)}{\lambda} \right] \right\}^{1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left\{ \log \left[ 1 - \frac {\log (1 - v)}{\lambda} \right] \right\}^{1 / b} dv \end{array}

for x > 0, 0 < p < 1, b > 0, the shape parameter, and \lambda > 0, the scale parameter.

Usage

dchen(x, b=1, lambda=1, log=FALSE)
pchen(x, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varchen(p, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
eschen(p, b=1, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dchen(x)
pchen(x)
varchen(x)
eschen(x)

Compound Laplace gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the compound Laplace gamma distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b}{2} \left\{ 1 + b \left | x - \theta \right | \right\}^{-\left( a + 1 \right)}, \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} \left\{ 1 + b \left | x - \theta \right | \right\}^{-a}, & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle 1 - \frac {1}{2} \left\{ 1 + b \left | x - \theta \right | \right\}^{-a}, & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \frac {1}{b} - \frac {(2 p)^{-1/a}}{b}, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta - \frac {1}{b} + \frac {(2 (1 - p))^{-1/a}}{b}, & \mbox{if $p > 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \frac {1}{b} - \frac {(2 p)^{-1/a}}{b (1 - 1/a)}, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta - \frac {1}{b} - \frac {\left[ 2 (1 - p) \right]^{1 - 1/a}}{2 p b (1 - 1/a)}, & \mbox{if $p > 1/2$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, b > 0, the scale parameter, and a > 0, the shape parameter.

Usage

dclg(x, a=1, b=1, theta=0, log=FALSE)
pclg(x, a=1, b=1, theta=0, log.p=FALSE, lower.tail=TRUE)
varclg(p, a=1, b=1, theta=0, log.p=FALSE, lower.tail=TRUE)
esclg(p, a=1, b=1, theta=0)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dclg(x)
pclg(x)
varclg(x)
esclg(x)

Complementary beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the complementary beta distribution due to Jones (2002) given by

\begin{array}{ll} &\displaystyle f (x) = B (a, b) \left\{ I_x^{-1} (a, b) \right\}^{1 - a} \left\{ 1 - I_x^{-1} (a, b) \right\}^{1 - b}, \\ &\displaystyle F (x) = I_x^{-1} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = I_p (a, b), \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p I_v (a, b) dv \end{array}

for 0 < x < 1, 0 < p < 1, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dcompbeta(x, a=1, b=1, log=FALSE)
pcompbeta(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varcompbeta(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
escompbeta(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dcompbeta(x)
pcompbeta(x)
varcompbeta(x)
escompbeta(x)

Dagum distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Dagum distribution due to Dagum (1975, 1977, 1980) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a c b^a x^{a c - 1}}{\left[ x^a + b^a \right]^{c + 1}}, \\ &\displaystyle F (x) = \left[ 1 + \left( \frac {b}{x} \right)^a \right]^{-c}, \\ &\displaystyle {\rm VaR}_p (X) = b \left( 1- p^{-1 / c} \right)^{-1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b}{p} \int_0^p \left( 1 - v^{-1 / c} \right)^{-1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, and c > 0, the second shape parameter.

Usage

ddagum(x, a=1, b=1, c=1, log=FALSE)
pdagum(x, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
vardagum(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
esdagum(p, a=1, b=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
ddagum(x)
pdagum(x)
vardagum(x)
esdagum(x)

Double Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the double Weibull distribution due to Balakrishnan and Kocherlakota (1985) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {c}{\displaystyle 2 \sigma} \left | \frac {x - \mu}{\sigma} \right |^{c - 1} \exp \left\{ -\left | \frac {x - \mu}{\sigma} \right |^c \right\}, \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} \exp \left\{ -\left( \frac {\mu - x}{\sigma} \right)^c \right\}, & \mbox{if $x \leq \mu$,} \\ \\ \displaystyle 1 - \frac {1}{2} \exp \left\{ -\left( \frac {x - \mu}{\sigma} \right)^c \right\}, & \mbox{if $x > \mu$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \mu - \sigma \left[ -\log \left( 2 p \right) \right]^{1 / c}, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \mu + \sigma \left[ -\log \left( 2 (1 - p) \right) \right]^{1 / c}, & \mbox{if $p > 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \mu - \frac {\sigma}{p} \int_0^p \left[ -\log 2 - \log v \right]^{1 / c} dv, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \mu - \frac {\sigma}{p} \int_0^{1/2} \left[ -\log 2 - \log v \right]^{1 / c} dv \\ \quad \displaystyle + \frac {\sigma}{p} \int_{1/2}^p \left[ -\log 2 - \log (1 - v) \right]^{1 / c} dv, & \mbox{if $p > 1/2$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, and c > 0, the shape parameter.

Usage

ddweibull(x, c=1, mu=0, sigma=1, log=FALSE)
pdweibull(x, c=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
vardweibull(p, c=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esdweibull(p, c=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
ddweibull(x)
pdweibull(x)
vardweibull(x)
esdweibull(x)

Exponentiated exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated exponential distribution due to Gupta and Kundu (1999, 2001) given by

\begin{array}{ll} &\displaystyle f (x) = a \lambda \exp (-\lambda x) \left[ 1 - \exp (-\lambda x) \right]^{a - 1}, \\ &\displaystyle F (x) = \left[ 1 - \exp (-\lambda x) \right]^{a}, \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{\lambda} \log \left( 1 - p^{1 / a} \right), \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{p \lambda} \int_0^p \log \left( 1 - v^{1 / a} \right) dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter and \lambda > 0, the scale parameter.

Usage

dexpexp(x, lambda=1, a=1, log=FALSE)
pexpexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
varexpexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
esexpexp(p, lambda=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexpexp(x)
pexpexp(x)
varexpexp(x)
esexpexp(x)

Exponential extension distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential extension distribution due to Nadarajah and Haghighi (2011) given by

\begin{array}{ll} &\displaystyle f (x) = a \lambda (1 + \lambda x)^{a - 1} \exp \left[ 1 - (1 + \lambda x)^a \right], \\ &\displaystyle F (x) = 1 - \exp \left[ 1 - (1 + \lambda x)^a \right], \\ &\displaystyle {\rm VaR}_p (X) = \frac {\left[ 1 - \log (1 - p) \right]^{1 / a} - 1}{\lambda}, \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{\lambda} + \frac {1}{\lambda p} \int_0^p \left[ 1 - \log (1 - v) \right]^{1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter and \lambda > 0, the scale parameter.

Usage

dexpext(x, lambda=1, a=1, log=FALSE)
pexpext(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
varexpext(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
esexpext(p, lambda=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexpext(x)
pexpext(x)
varexpext(x)
esexpext(x)

Exponential geometric distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential geometric distribution due to Adamidis and Loukas (1998) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {\lambda \theta \exp (-\lambda x)}{\left[ 1 - (1 - \theta) \exp (-\lambda x) \right]^2}, \\ &\displaystyle F (x) = \frac {\theta \exp (-\lambda x)}{1 - (1 - \theta) \exp (-\lambda x)}, \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{\lambda} \log \frac {p}{\theta + (1 - \theta) p}, \\ &\displaystyle {\rm ES}_p (X) = -\frac {\log p}{\lambda} - \frac {\theta \log \theta}{\lambda p (1 - \theta)} + \frac {\theta + (1 - \theta) p}{\lambda p (1 - \theta)} \log \left[ \theta + (1 - \theta) p \right] \end{array}

for x > 0, 0 < p < 1, 0 < \theta < 1, the first scale parameter, and \lambda > 0, the second scale parameter.

Usage

dexpgeo(x, theta=0.5, lambda=1, log=FALSE)
pexpgeo(x, theta=0.5, lambda=1, log.p=FALSE, lower.tail=TRUE)
varexpgeo(p, theta=0.5, lambda=1, log.p=FALSE, lower.tail=TRUE)
esexpgeo(p, theta=0.5, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexpgeo(x)
pexpgeo(x)
varexpgeo(x)
esexpgeo(x)

Exponential logarithmic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential logarithmic distribution due to Tahmasbi and Rezaei (2008) given by

\begin{array}{ll} &\displaystyle f(x) = -\frac {b (1 - a) \exp (-b x)}{\log a \left[ 1 - (1 - a) \exp (-b x) \right]}, \\ &\displaystyle F (x) = 1 - \frac {\log \left[ 1 - (1 - a) \exp (-b x) \right]}{\log a}, \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{b} \log \left[ \frac {1 - a^{1 - p}}{1 - a} \right], \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{b p} \int_0^p \log \left[ \frac {1 - a^{1 - v}}{1 - a} \right] dv \end{array}

for x > 0, 0 < p < 1, 0 < a < 1, the first scale parameter, and b > 0, the second scale parameter.

Usage

dexplog(x, a=0.5, b=1, log=FALSE)
pexplog(x, a=0.5, b=1, log.p=FALSE, lower.tail=TRUE)
varexplog(p, a=0.5, b=1, log.p=FALSE, lower.tail=TRUE)
esexplog(p, a=0.5, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexplog(x)
pexplog(x)
varexplog(x)
esexplog(x)

Exponentiated logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated logistic distribution given by

\begin{array}{ll} &\displaystyle f (x) = (a/b) \exp (-x/b) \left[ 1 + \exp (-x/b) \right]^{-a - 1}, \\ &\displaystyle F (x) = \left[ 1 + \exp (-x/b) \right]^{-a}, \\ &\displaystyle {\rm VaR}_p (X) = -b \log \left[ p^{-1 / a} - 1 \right], \\ &\displaystyle {\rm ES}_p (X) = -\frac {b}{p} \int_0^p \log \left[ v^{-1 / a} - 1 \right] dv \end{array}

for -\infty < x < \infty, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.

Usage

dexplogis(x, a=1, b=1, log=FALSE)
pexplogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varexplogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esexplogis(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexplogis(x)
pexplogis(x)
varexplogis(x)
esexplogis(x)

Exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential distribution given by

\begin{array}{ll} &\displaystyle f(x) = \lambda \exp (-\lambda x), \\ &\displaystyle F (x) = 1 - \exp (-\lambda x), \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{\lambda} \log (1 - p), \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{p \lambda} \left\{ \log (1 - p) p - p - \log (1 - p) \right\} \end{array}

for x > 0, 0 < p < 1, and \lambda > 0, the scale parameter.

Usage

dexponential(x, lambda=1, log=FALSE)
pexponential(x, lambda=1, log.p=FALSE, lower.tail=TRUE)
varexponential(p, lambda=1, log.p=FALSE, lower.tail=TRUE)
esexponential(p, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexponential(x)
pexponential(x)
varexponential(x)
esexponential(x)

Exponential Poisson distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential Poisson distribution due to Kus (2007) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {b \lambda \exp \left[ -b x - \lambda + \lambda \exp (-b x) \right]}{1 - \exp (-\lambda)}, \\ &\displaystyle F (x) = \frac {1 - \exp \left[ -\lambda + \lambda \exp (-b x) \right]}{1 - \exp (-\lambda)}, \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{b} \log \left\{ \frac {1}{\lambda} \log \left[ 1 - p + p \exp (-\lambda) \right] + 1 \right\}, \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{b p} \int_0^p \log \left\{ \frac {1}{\lambda} \log \left[ 1 - v + v \exp (-\lambda) \right] + 1 \right\} dv \end{array}

for x > 0, 0 < p < 1, b > 0, the first scale parameter, and \lambda > 0, the second scale parameter.

Usage

dexppois(x, b=1, lambda=1, log=FALSE)
pexppois(x, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varexppois(p, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
esexppois(p, b=1, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexppois(x)
pexppois(x)
varexppois(x)
esexppois(x)

Exponential power distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential power distribution due to Subbotin (1923) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\displaystyle 2 a^{1/a} \sigma \Gamma \left( 1 + 1/a \right)} \exp \left\{ -\frac {\mid x - \mu \mid^a}{a \sigma^a} \right\}, \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} Q \left( \frac {1}{a}, \frac {(\mu - x)^a}{a \sigma^a} \right), & \mbox{if $x \leq \mu$,} \\ \\ \displaystyle 1 - \frac {1}{2} Q \left( \frac {1}{a}, \frac {(x - \mu)^a}{a \sigma^a} \right), & \mbox{if $x > \mu$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \mu - a^{1/a} \sigma \left[ Q^{-1} \left( \frac {1}{a}, 2 p \right) \right]^{1/a}, & \mbox{if $p \leq 1/2$,} \\ \\ \mu + a^{1/a} \sigma \left[ Q^{-1} \left( \frac {1}{a}, 2 (1 - p) \right) \right]^{1/a}, & \mbox{if $p > 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \mu - \frac {a^{1/a} \sigma}{p} \int_0^p \left[ Q^{-1} \left( \frac {1}{a}, 2 v \right) \right]^{1/a} dv, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \mu - \frac {a^{1/a} \sigma}{p} \int_0^{1/2} \left[ Q^{-1} \left( \frac {1}{a}, 2 v \right) \right]^{1/a} dv \\ \displaystyle \quad +\frac {a^{1/a} \sigma}{p} \int_{1/2}^p \left[ Q^{-1} \left( \frac {1}{a}, 2 (1 - v) \right) \right]^{1/a} dv, & \mbox{if $p > 1/2$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, and a > 0, the shape parameter.

Usage

dexppower(x, mu=0, sigma=1, a=1, log=FALSE)
pexppower(x, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)
varexppower(p, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)
esexppower(p, mu=0, sigma=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexppower(x)
pexppower(x)
varexppower(x)
esexppower(x)

Exponentiated Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated Weibull distribution due to Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) given by

\begin{array}{ll} &\displaystyle f(x) = a \alpha \sigma^{-\alpha} x^{\alpha - 1} \exp \left[ -(x / \sigma)^\alpha \right] \left\{ 1 - \exp \left[ -(x / \sigma)^\alpha \right] \right\}^{a - 1}, \\ &\displaystyle F (x) = \left\{ 1 - \exp \left[ -(x / \sigma)^\alpha \right] \right\}^a, \\ &\displaystyle {\rm VaR}_p (X) = \sigma \left[ -\log \left( 1 - p^{1 / a} \right) \right]^{1 / \alpha}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left[ -\log \left( 1 - v^{1 / a} \right) \right]^{1 / \alpha} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, \alpha > 0, the second shape parameter, and \sigma > 0, the scale parameter.

Usage

dexpweibull(x, a=1, alpha=1, sigma=1, log=FALSE)
pexpweibull(x, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varexpweibull(p, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esexpweibull(p, a=1, alpha=1, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dexpweibull(x)
pexpweibull(x)
varexpweibull(x)
esexpweibull(x)

Frechet distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Fr\'echet distribution due to Fr\'echet (1927) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\alpha \sigma^\alpha}{x^{\alpha + 1}} \exp \left\{ -\left( \frac {\sigma}{x} \right)^{\alpha} \right\}, \\ &\displaystyle F (x) = \exp \left\{ -\left( \frac {\sigma}{x} \right)^{\alpha} \right\}, \\ &\displaystyle {\rm VaR}_p (X) = \sigma \left[ -\log p \right]^{-1 / \alpha}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \Gamma \left( 1 - 1 / \alpha, -\log p \right) \end{array}

for x > 0, 0 < p < 1, \alpha > 0, the shape parameter, and \sigma > 0, the scale parameter, where \Gamma (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt denotes the complementary incomplete gamma function.

Usage

dfrechet(x, alpha=1, sigma=1, log=FALSE)
pfrechet(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varfrechet(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esfrechet(p, alpha=1, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dfrechet(x)
pfrechet(x)
varfrechet(x)
esfrechet(x)

Generalized beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {(x - c)^{a - 1} (d - x)^{b - 1}}{B (a, b) (d - c)^{a + b - 1}}, \\ &\displaystyle F (x) = I_{\frac {x - c}{d - c}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = c + (d - c) I_p^{-1} (a, b), \\ &\displaystyle {\rm ES}_p (X) = c + \frac {d - c}{p} \int_0^p I_v^{-1} (a, b) dv \end{array}

for c \leq x \leq d, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, -\infty < c < \infty, the first location parameter, and -\infty < c < d < \infty, the second location parameter.

Usage

dgenbeta(x, a=1, b=1, c=0, d=1, log=FALSE)
pgenbeta(x, a=1, b=1, c=0, d=1, log.p=FALSE, lower.tail=TRUE)
vargenbeta(p, a=1, b=1, c=0, d=1, log.p=FALSE, lower.tail=TRUE)
esgenbeta(p, a=1, b=1, c=0, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenbeta(x)
pgenbeta(x)
vargenbeta(x)
esgenbeta(x)

Generalized beta II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta II distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {c x^{ac - 1} \left( 1 - x^c \right)^{b - 1}}{B (a, b)}, \\ &\displaystyle F (x) = I_{x^c} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \left[ I_p^{-1} (a, b) \right]^{1 / c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ I_v^{-1} (a, b) \right]^{1 / c} dv \end{array}

for 0 < x < 1, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, and c > 0, the third shape parameter.

Usage

dgenbeta2(x, a=1, b=1, c=1, log=FALSE)
pgenbeta2(x, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
vargenbeta2(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
esgenbeta2(p, a=1, b=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenbeta2(x)
pgenbeta2(x)
vargenbeta2(x)
esgenbeta2(x)

Generalized inverse beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized inverse beta distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a x^{ac - 1}}{B (c, d) \left( 1 + x^a \right)^{c + d}}, \\ &\displaystyle F (x) = I_{\frac {x^a}{1 + x^a}} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = \left[ \frac {I_p^{-1} (c, d)}{1 - I_p^{-1} (c, d)} \right]^{1/a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ \frac {I_v^{-1} (c, d)}{1 - I_v^{-1} (c, d)} \right]^{1/a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter.

Usage

dgeninvbeta(x, a=1, c=1, d=1, log=FALSE)
pgeninvbeta(x, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
vargeninvbeta(p, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
esgeninvbeta(p, a=1, c=1, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgeninvbeta(x)
pgeninvbeta(x)
vargeninvbeta(x)
esgeninvbeta(x)

Generalized logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a \exp \left( -\frac {x - \mu}{\sigma} \right)} {\sigma \left\{ 1 + \exp \left( -\frac {x - \mu}{\sigma} \right) \right\}^{1 + a}}, \\ &\displaystyle F (x) = \frac {1}{\left\{ 1 + \exp \left( -\frac {x - \mu}{\sigma} \right) \right\}^a}, \\ &\displaystyle {\rm VaR}_p (X) = \mu - \sigma \log \left( p^{-1 / a} - 1 \right), \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{p} \int_0^p \log \left( v^{-1 / a} - 1 \right) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, and a > 0, the shape parameter.

Usage

dgenlogis(x, a=1, mu=0, sigma=1, log=FALSE)
pgenlogis(x, a=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
vargenlogis(p, a=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esgenlogis(p, a=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenlogis(x)
pgenlogis(x)
vargenlogis(x)
esgenlogis(x)

Generalized logistic III distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic III distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma B (\alpha, \alpha)} \exp \left( \alpha \frac {x - \mu}{\sigma} \right) \left\{ 1 + \exp \left( \frac {x - \mu}{\sigma} \right) \right\}^{-2 \alpha}, \\ &\displaystyle F (x) = I_{\frac {1}{1 + \exp \left( -\frac {x - \mu}{\sigma} \right)}} (\alpha, \alpha), \\ &\displaystyle {\rm VaR}_p (X) = \mu - \sigma \log \frac {1 - I_p^{-1} (\alpha, \alpha)}{I_p^{-1} (\alpha, \alpha)}, \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{p} \int_0^p \log \frac {1 - I_v^{-1} (\alpha, \alpha)}{I_v^{-1} (\alpha, \alpha)} dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, and \alpha > 0, the shape parameter.

Usage

dgenlogis3(x, alpha=1, mu=0, sigma=1, log=FALSE)
pgenlogis3(x, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
vargenlogis3(p, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esgenlogis3(p, alpha=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenlogis3(x)
pgenlogis3(x)
vargenlogis3(x)
esgenlogis3(x)

Generalized logistic IV distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic IV distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma B (\alpha, a)} \exp \left( -\alpha \frac {x - \mu}{\sigma} \right) \left\{ 1 + \exp \left( -\frac {x - \mu}{\sigma} \right) \right\}^{-\alpha - a}, \\ &\displaystyle F (x) = I_{\frac {1}{1 + \exp \left( -\frac {x - \mu}{\sigma} \right)}} (\alpha, a), \\ &\displaystyle {\rm VaR}_p (X) = \mu - \sigma \log \frac {1 - I_p^{-1} (\alpha, a)}{I_p^{-1} (\alpha, a)}, \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{p} \int_0^p \log \frac {1 - I_v^{-1} (\alpha, a)}{I_v^{-1} (\alpha, a)} dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, \alpha > 0, the first shape parameter, and a > 0, the second shape parameter.

Usage

dgenlogis4(x, a=1, alpha=1, mu=0, sigma=1, log=FALSE)
pgenlogis4(x, a=1, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
vargenlogis4(p, a=1, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esgenlogis4(p, a=1, alpha=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenlogis4(x)
pgenlogis4(x)
vargenlogis4(x)
esgenlogis4(x)

Generalized Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized Pareto distribution due to Pickands (1975) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{k} \left( 1 - \frac {c x}{k} \right)^{1 / c - 1}, \\ &\displaystyle F (x) = 1 - \left( 1 - \frac {c x}{k} \right)^{1 / c}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {k}{c} \left[ 1 - (1 - p)^c \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {k}{c} + \frac {k (1 - p)^{c + 1}}{p c (c + 1)} - \frac {k}{p c (c + 1)} \end{array}

for x < k/c if c > 0, x > k/c if c < 0, x > 0 if c = 0, 0 < p < 1, k > 0, the scale parameter and -\infty < c < \infty, the shape parameter.

Usage

dgenpareto(x, k=1, c=1, log=FALSE)
pgenpareto(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
vargenpareto(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
esgenpareto(p, k=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenpareto(x)
pgenpareto(x)
vargenpareto(x)
esgenpareto(x)

Generalized power Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized power Weibull distribution due to Nikulin and Haghighi (2006) given by

\begin{array}{ll} &\displaystyle f(x) = a \theta x^{a - 1} \left[ 1 + x^a \right]^{\theta - 1} \exp \left\{ 1 - \left[ 1 + x^a \right]^\theta \right\}, \\ &\displaystyle F (x) = 1 - \exp \left\{ 1 - \left[ 1 + x^a \right]^\theta \right\}, \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \left[ 1 - \log (1 - p) \right]^{1 / \theta} - 1 \right\}^{1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left\{ \left[ 1 - \log (1 - v) \right]^{1 / \theta} - 1 \right\}^{1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, and \theta > 0, the second shape parameter.

Usage

dgenpowerweibull(x, a=1, theta=1, log=FALSE)
pgenpowerweibull(x, a=1, theta=1, log.p=FALSE, lower.tail=TRUE)
vargenpowerweibull(p, a=1, theta=1, log.p=FALSE, lower.tail=TRUE)
esgenpowerweibull(p, a=1, theta=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenpowerweibull(x)
pgenpowerweibull(x)
vargenpowerweibull(x)
esgenpowerweibull(x)

Generalized uniform distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized uniform distribution given by

\begin{array}{ll} &\displaystyle f (x) = h k c (x - a)^{c - 1} \left[ 1 - k (x - a)^c \right]^{h - 1}, \\ &\displaystyle F (x) = 1 - \left[ 1 - k (x - a)^c \right]^h, \\ &\displaystyle {\rm VaR}_p (X) = a + k^{-1/c} \left[ 1 - (1 - p)^{1/h} \right]^{1/c}, \\ &\displaystyle {\rm ES}_p (X) = a + \frac {k^{-1/c}}{p} \int_0^p \left[ 1 - (1 - v)^{1/h} \right]^{1/c} dv \end{array}

for a \leq x \leq a + k^{-1/c}, 0 < p < 1, -\infty < a < \infty, the location parameter, c > 0, the first shape parameter, k > 0, the scale parameter, and h > 0, the second shape parameter.

Usage

dgenunif(x, a=0, c=1, h=1, k=1, log=FALSE)
pgenunif(x, a=0, c=1, h=1, k=1, log.p=FALSE, lower.tail=TRUE)
vargenunif(p, a=0, c=1, h=1, k=1, log.p=FALSE, lower.tail=TRUE)
esgenunif(p, a=0, c=1, h=1, k=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgenunif(x)
pgenunif(x)
vargenunif(x)
esgenunif(x)

Generalized extreme value distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized extreme value distribution due to Fisher and Tippett (1928) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {1}{\sigma} \left[ 1 + \xi \left( \frac {x - \mu}{\sigma} \right) \right]^{-1/\xi - 1} \exp \left\{ -\left[ 1 + \xi \left( \frac {x - \mu}{\sigma} \right) \right]^{-1/\xi} \right\}, \\ &\displaystyle F(x) = \exp \left\{ -\left[ 1 + \xi \left( \frac {x - \mu}{\sigma} \right) \right]^{-1/\xi} \right\}, \\ &\displaystyle {\rm VaR}_p (X) = \mu - \frac {\sigma}{\xi} + \frac {\sigma}{\xi} (-\log p)^{-\xi}, \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{\xi} + \frac {\sigma}{p \xi} \int_0^p (-\log v)^{-\xi} dv \end{array}

for x \geq \mu - \sigma / \xi if \xi > 0, x \leq \mu - \sigma / \xi if \xi < 0, -\infty < x < \infty if \xi = 0, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, and -\infty < \xi < \infty, the shape parameter.

Usage

dgev(x, mu=0, sigma=1, xi=1, log=FALSE)
pgev(x, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)
vargev(p, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)
esgev(p, mu=0, sigma=1, xi=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgev(x)
pgev(x)
vargev(x)
esgev(x)

Gompertz distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gompertz distribution due to Gompertz (1825) given by

\begin{array}{ll} &\displaystyle f(x) = b \eta \exp (bx) \exp \left[ \eta - \eta \exp (bx) \right], \\ &\displaystyle F (x) = 1 - \exp \left[ \eta - \eta \exp (bx) \right], \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} \log \left[ 1 - \frac {1}{\eta} \log (1 - p) \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p b} \int_0^p \log \left[ 1 - \frac {1}{\eta} \log (1 - v) \right] dv \end{array}

for x > 0, 0 < p < 1, b > 0, the first scale parameter and \eta > 0, the second scale parameter.

Usage

dgompertz(x, b=1, eta=1, log=FALSE)
pgompertz(x, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)
vargompertz(p, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)
esgompertz(p, b=1, eta=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgompertz(x)
pgompertz(x)
vargompertz(x)
esgompertz(x)

Gumbel distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel distribution given by due to Gumbel (1954) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma} \exp \left( \frac {\mu - x}{\sigma} \right) \exp \left[ -\exp \left( \frac {\mu - x}{\sigma} \right) \right], \\ &\displaystyle F (x) = \exp \left[ -\exp \left( \frac {\mu - x}{\sigma} \right) \right], \\ &\displaystyle {\rm VaR}_p (X) = \mu - \sigma \log (-\log p), \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{p} \int_0^p \log (-\log v) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter.

Usage

dgumbel(x, mu=0, sigma=1, log=FALSE)
pgumbel(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
vargumbel(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esgumbel(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgumbel(x)
pgumbel(x)
vargumbel(x)
esgumbel(x)

Gumbel II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel II distribution

\begin{array}{ll} &\displaystyle f (x) = a b x^{-a - 1} \exp \left( -b x^{-a} \right), \\ &\displaystyle F (x) = 1 - \exp \left( -b x^{-a} \right), \\ &\displaystyle {\rm VaR}_p (X) = b^{1 / a} \left[ -\log (1 - p) \right]^{-1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b^{1 / a}}{p} \int_0^p \left[ -\log (1 - v) \right]^{-1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.

Usage

dgumbel2(x, a=1, b=1, log=FALSE)
pgumbel2(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
vargumbel2(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esgumbel2(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dgumbel2(x)
pgumbel2(x)
vargumbel2(x)
esgumbel2(x)

Half Student's t distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the half Student's t distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {2 \Gamma \left( \frac {n + 1}{2} \right)}{\sqrt{n \pi} \Gamma \left( \frac {n}{2} \right)} \left( 1 + \frac {x^2}{n} \right)^{-\frac {n + 1}{2}}, \\ &\displaystyle F (x) = I_{\frac {x^2}{x^2 + n}} \left( \frac {1}{2}, \frac {n}{2} \right), \\ &\displaystyle {\rm VaR}_p (X) = \sqrt{\frac {n I_p^{-1} \left( \frac {1}{2}, \frac {n}{2} \right)} {1 - I_p^{-1} \left( \frac {1}{2}, \frac {n}{2} \right)}}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sqrt{n}}{p} \int_0^p \sqrt{\frac {I_v^{-1} \left( \frac {1}{2}, \frac {n}{2} \right)} {1 - I_v^{-1} \left( \frac {1}{2}, \frac {n}{2} \right)}} dv \end{array}

for -\infty < x < \infty, 0 < p < 1, and n > 0, the degree of freedom parameter.

Usage

dhalfT(x, n=1, log=FALSE)
phalfT(x, n=1, log.p=FALSE, lower.tail=TRUE)
varhalfT(p, n=1, log.p=FALSE, lower.tail=TRUE)
eshalfT(p, n=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dhalfT(x)
phalfT(x)
varhalfT(x)
eshalfT(x)

Half Cauchy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the half Cauchy distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {2}{\pi} \frac {\sigma}{x^2 + \sigma^2}, \\ &\displaystyle F (x) = \frac {2}{\pi} \arctan \left( \frac {x}{\sigma} \right), \\ &\displaystyle {\rm VaR}_p (X) = \sigma \tan \left( \frac {\pi p}{2} \right), \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \tan \left( \frac {\pi v}{2} \right) dv \end{array}

for x > 0, 0 < p < 1, and \sigma > 0, the scale parameter.

Usage

dhalfcauchy(x, sigma=1, log=FALSE)
phalfcauchy(x, sigma=1, log.p=FALSE, lower.tail=TRUE)
varhalfcauchy(p, sigma=1, log.p=FALSE, lower.tail=TRUE)
eshalfcauchy(p, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dhalfcauchy(x)
phalfcauchy(x)
varhalfcauchy(x)
eshalfcauchy(x)

Half logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the half logistic distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {2 \lambda \exp \left( -\lambda x \right)} {\left[ 1 + \exp \left( -\lambda x \right) \right]^2}, \\ &\displaystyle F (x) = \frac {1 - \exp \left( -\lambda x \right)}{1 + \exp \left( -\lambda x \right)}, \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{\lambda} \log \frac {1 - p}{1 + p}, \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{\lambda} \log \frac {1 - p}{1 + p} + \frac {1}{\lambda p} \log \left( 1 - p^2 \right) \end{array}

for x > 0, 0 < p < 1, and \lambda > 0, the scale parameter.

Usage

dhalflogis(x, lambda=1, log=FALSE)
phalflogis(x, lambda=1, log.p=FALSE, lower.tail=TRUE)
varhalflogis(p, lambda=1, log.p=FALSE, lower.tail=TRUE)
eshalflogis(p, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dhalflogis(x)
phalflogis(x)
varhalflogis(x)
eshalflogis(x)

Half normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for Half normal distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {2}{\sigma} \phi \left( \frac {x}{\sigma} \right), \\ &\displaystyle F (x) = 2 \Phi \left( \frac {x}{\sigma} \right) - 1, \\ &\displaystyle {\rm VaR}_p (X) = \sigma \Phi^{-1} \left( \frac {1 + p}{2} \right), \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \Phi^{-1} \left( \frac {1 + v}{2} \right) dv \end{array}

for x > 0, 0 < p < 1, and \sigma > 0, the scale parameter.

Usage

dhalfnorm(x, sigma=1, log=FALSE)
phalfnorm(x, sigma=1, log.p=FALSE, lower.tail=TRUE)
varhalfnorm(p, sigma=1, log.p=FALSE, lower.tail=TRUE)
eshalfnorm(p, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dhalfnorm(x)
phalfnorm(x)
varhalfnorm(x)
eshalfnorm(x)

Inverse beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the inverse beta distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {x^{a - 1}}{B (a, b) (1 + x)^{a + b}}, \\ &\displaystyle F (x) = I_{\frac {x}{1 + x}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = \frac {I_p^{-1} (a, b)}{1 - I_p^{-1} (a, b)}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \frac {I_v^{-1} (a, b)}{1 - I_v^{-1} (a, b)} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dinvbeta(x, a=1, b=1, log=FALSE)
pinvbeta(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varinvbeta(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esinvbeta(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dinvbeta(x)
pinvbeta(x)
varinvbeta(x)
esinvbeta(x)

Inverse exponentiated exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the inverse exponentiated exponential distribution due to Ghitany et al. (2013) given by

\begin{array}{ll} &\displaystyle f (x) = a \lambda x^{-2} \exp \left(-\frac {\lambda}{x} \right) \left[ 1 - \exp \left( -\frac {\lambda}{x} \right) \right]^{a - 1}, \\ &\displaystyle F (x) = 1 - \left[ 1 - \exp \left( -\frac {\lambda}{x} \right) \right]^a, \\ &\displaystyle {\rm VaR}_p (X) = \lambda \left\{ -\log \left[ 1 - (1 - p)^{1 / a} \right] \right\}^{-1}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\lambda}{p} \int_0^p \left\{ -\log \left[ 1 - (1 - v)^{1 / a} \right] \right\}^{-1} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter and \lambda > 0, the scale parameter.

Usage

dinvexpexp(x, lambda=1, a=1, log=FALSE)
pinvexpexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
varinvexpexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
esinvexpexp(p, lambda=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dinvexpexp(x)
pinvexpexp(x)
varinvexpexp(x)
esinvexpexp(x)

Inverse gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the inverse gamma distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b^a \exp (-b / x)}{x^{a + 1} \Gamma (a)}, \\ &\displaystyle F (x) = Q (a, b / x), \\ &\displaystyle {\rm VaR}_p (X) = b \left[ Q^{-1} (a, p) \right]^{-1}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b}{p} \int_0^p \left[ Q^{-1} (a, v) \right]^{-1} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.

Usage

dinvgamma(x, a=1, b=1, log=FALSE)
pinvgamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varinvgamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esinvgamma(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dinvgamma(x)
pinvgamma(x)
varinvgamma(x)
esinvgamma(x)

Kumaraswamy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy distribution due to Kumaraswamy (1980) given by

\begin{array}{ll} &\displaystyle f (x) = a b x^{a - 1} \left( 1 - x^a \right)^{b - 1}, \\ &\displaystyle F (x) = 1 - \left( 1 - x^a \right)^b, \\ &\displaystyle {\rm VaR}_p (X) = \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} dv \end{array}

for 0 < x < 1, 0 < p < 1, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dkum(x, a=1, b=1, log=FALSE)
pkum(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkum(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskum(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkum(x)
pkum(x)
varkum(x)
eskum(x)

Kumaraswamy Burr XII distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Burr XII distribution due to Parana\'iba et al. (2013) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b k c x^{c - 1}}{\left( 1 + x^c \right)^{k + 1}} \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^{a - 1} \left\{ 1 - \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^a \right\}^{b - 1}, \\ &\displaystyle F (x) = 1 - \left\{ 1 - \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^a \right\}^b, \\ &\displaystyle {\rm VaR}_p (X) = \left[ \left\{ 1 - \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\}^{-1 / k} - 1 \right]^{1/c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ \left\{ 1 - \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\}^{-1 / k} - 1 \right]^{1/c} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, c > 0, the third shape parameter, and k > 0, the fourth shape parameter.

Usage

dkumburr7(x, a=1, b=1, k=1, c=1, log=FALSE)
pkumburr7(x, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
varkumburr7(p, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
eskumburr7(p, a=1, b=1, k=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumburr7(x)
pkumburr7(x)
varkumburr7(x)
eskumburr7(x)

Kumaraswamy exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy exponential distribution due to Cordeiro and de Castro (2011) given by

\begin{array}{ll} &\displaystyle f (x) = a b \lambda \exp (-\lambda x) \left[ 1 - \exp (-\lambda x) \right]^{a - 1} \left\{ 1 - \left[ 1 - \exp (-\lambda x) \right]^a \right\}^{b - 1}, \\ &\displaystyle F (x) = 1 - \left\{ 1 - \left[ 1 - \exp (-\lambda x) \right]^a \right\}^b, \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{\lambda} \log \left\{ 1 - \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\}, \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{p \lambda} \int_0^p \log \left\{ 1 - \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, and \lambda > 0, the scale parameter.

Usage

dkumexp(x, lambda=1, a=1, b=1, log=FALSE)
pkumexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkumexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskumexp(p, lambda=1, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumexp(x)
pkumexp(x)
varkumexp(x)
eskumexp(x)

Kumaraswamy gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy gamma distribution due to de Pascoa et al. (2011) given by

\begin{array}{ll} &\displaystyle f (x) = c d b^a x^{a - 1} \exp (-b x) \frac {\gamma^{c - 1} (a, b x)}{\Gamma^c (a)} \left[ 1 - \frac {\gamma^c (a, b x)}{\Gamma^c (a)} \right]^{d - 1}, \\ &\displaystyle F (x) = 1 - \left[ 1 - \frac {\gamma^c (a, b x)}{\Gamma^c (a)} \right]^d, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} Q^{-1} \left( a, 1 - \left[ 1 - (1 - p)^{1 / d} \right]^{1 / c} \right), \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{b p} \int_0^p Q^{-1} \left( a, 1 - \left[ 1 - (1 - v)^{1 / d} \right]^{1 / c} \right) dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter.

Usage

dkumgamma(x, a=1, b=1, c=1, d=1, log=FALSE)
pkumgamma(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
varkumgamma(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
eskumgamma(p, a=1, b=1, c=1, d=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumgamma(x)
pkumgamma(x)
varkumgamma(x)
eskumgamma(x)

Kumaraswamy Gumbel distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Gumbel distribution due to Cordeiro et al. (2012a) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b}{\sigma} \exp \left( \frac {\mu - x}{\sigma} \right) \exp \left[ -a \exp \left( \frac {\mu - x}{\sigma} \right) \right] \left\{ 1 - \exp \left[ -a \exp \left( \frac {\mu - x}{\sigma} \right) \right] \right\}^{b - 1}, \\ &\displaystyle F (x) = 1 - \left\{ 1 - \exp \left[ -a \exp \left( \frac {\mu - x}{\sigma} \right) \right] \right\}^b, \\ &\displaystyle {\rm VaR}_p (X) = \mu - \sigma \log \left\{ -\log \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\}, \\ &\displaystyle {\rm ES}_p (X) = \mu - \frac {\sigma}{p} \int_0^p \log \left\{ -\log \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\} dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dkumgumbel(x, a=1, b=1, mu=0, sigma=1, log=FALSE)
pkumgumbel(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varkumgumbel(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eskumgumbel(p, a=1, b=1, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumgumbel(x)
pkumgumbel(x)
varkumgumbel(x)
eskumgumbel(x)

Kumaraswamy half normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy half normal distribution due to Cordeiro et al. (2012c) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {2a b}{\sigma} \phi \left( \frac {x}{\sigma} \right) \left[ 2 \Phi \left( \frac {x}{\sigma} \right) - 1 \right]^{a - 1} \left\{ 1 - \left[ 2 \Phi \left( \frac {x}{\sigma} \right) - 1 \right]^a \right\}^{b - 1}, \\ &\displaystyle F (x) = 1 - \left\{ 1 - \left[ 2 \Phi \left( \frac {x}{\sigma} \right) - 1 \right]^a \right\}^b, \\ &\displaystyle {\rm VaR}_p (X) = \sigma \Phi^{-1} \left( \frac {1}{2} + \frac {1}{2} \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right), \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \Phi^{-1} \left( \frac {1}{2} + \frac {1}{2} \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right) dv \end{array}

for x > 0, 0 < p < 1, \sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dkumhalfnorm(x, sigma=1, a=1, b=1, log=FALSE)
pkumhalfnorm(x, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkumhalfnorm(p, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskumhalfnorm(p, sigma=1, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumhalfnorm(x)
pkumhalfnorm(x)
varkumhalfnorm(x)
eskumhalfnorm(x)

Kumaraswamy log-logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy log-logistic distribution due to de Santana et al. (2012) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b \beta \alpha^\beta x^{a \beta - 1}} {\left( \alpha^\beta + x^\beta \right)^{a + 1}} \left[ 1 - \frac {x^{a \beta}}{\left( \alpha^\beta + x^\beta \right)^a} \right]^{b - 1}, \\ &\displaystyle F (x) = \left[ 1 - \frac {x^{a \beta}}{\left( \alpha^\beta + x^\beta \right)^a} \right]^b, \\ &\displaystyle {\rm VaR}_p (X) = \alpha \left\{ \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} - 1 \right\}^{-1 / \beta}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\alpha}{p} \int_0^p \left\{ \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} - 1 \right\}^{-1 / \beta} dv \end{array}

for x > 0, 0 < p < 1, \alpha > 0, the scale parameter, \beta > 0, the first shape parameter, a > 0, the second shape parameter, and b > 0, the third shape parameter.

Usage

dkumloglogis(x, a=1, b=1, alpha=1, beta=1, log=FALSE)
pkumloglogis(x, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)
varkumloglogis(p, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)
eskumloglogis(p, a=1, b=1, alpha=1, beta=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumloglogis(x)
pkumloglogis(x)
varkumloglogis(x)
eskumloglogis(x)

Kumaraswamy normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for Kumaraswamy normal distribution due to Cordeiro and de Castro (2011) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b}{\sigma} \phi \left( \frac {x - \mu}{\sigma} \right) \Phi^{a - 1} \left( \frac {x - \mu}{\sigma} \right) \left[ 1 - \Phi^a \left( \frac {x - \mu}{\sigma} \right) \right]^{b - 1}, \\ &\displaystyle F (x) = 1 - \left[ 1 - \Phi^a \left( \frac {x - \mu}{\sigma} \right) \right]^b, \\ &\displaystyle {\rm VaR}_p (X) = \mu + \sigma \Phi^{-1} \left( \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right), \\ &\displaystyle {\rm ES}_p (X) = \mu + \frac {\sigma}{p} \int_0^p \Phi^{-1} \left( \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, \sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dkumnormal(x, mu=0, sigma=1, a=1, b=1, log=FALSE)
pkumnormal(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkumnormal(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskumnormal(p, mu=0, sigma=1, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumnormal(x)
pkumnormal(x)
varkumnormal(x)
eskumnormal(x)

Kumaraswamy Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Pareto distribution due to Pereira et al. (2013) given by

\begin{array}{ll} &\displaystyle f (x) = a b c K^c x^{-c - 1} \left[ 1 - \left( \frac {K}{x} \right)^c \right]^{a - 1} \left\{ 1 - \left[ 1 - \left( \frac {K}{x} \right)^c \right]^a \right\}^{b - 1}, \\ &\displaystyle F (x) = 1 - \left\{ 1 - \left[ 1 - \left( \frac {K}{x} \right)^c \right]^a \right\}^b, \\ &\displaystyle {\rm VaR}_p (X) = K \left\{ 1 - \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\}^{-1 / c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {K}{p} \int_0^p \left\{ 1 - \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\}^{-1 / c} dv \end{array}

for x \geq K, 0 < p < 1, K > 0, the scale parameter, c > 0, the first shape parameter, a > 0, the second shape parameter, and b > 0, the third shape parameter.

Usage

dkumpareto(x, K=1, a=1, b=1, c=1, log=FALSE)
pkumpareto(x, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
varkumpareto(p, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
eskumpareto(p, K=1, a=1, b=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumpareto(x)
pkumpareto(x)
varkumpareto(x)
eskumpareto(x)

Kumaraswamy Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Weibull distribution due to Cordeiro et al. (2010) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b \alpha x^{\alpha - 1}}{\sigma^\alpha} \exp \left[ -\left( \frac {x}{\sigma} \right)^{\alpha} \right] \left\{ 1 - \exp \left[ -\left( \frac {x}{\sigma} \right)^{\alpha} \right] \right\}^{a - 1} \left[ 1 - \left\{ 1 - \exp \left[ -\left( \frac {x}{\sigma} \right)^{\alpha} \right] \right\}^a \right]^{b - 1}, \\ &\displaystyle F (x) = 1 - \left[ 1 - \left\{ 1 - \exp \left[ -\left( \frac {x}{\sigma} \right)^{\alpha} \right] \right\}^a \right]^b, \\ &\displaystyle {\rm VaR}_p (X) = \sigma \left[ -\log \left\{ 1 - \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\} \right]^{1 / \alpha}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left[ -\log \left\{ 1 - \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\} \right]^{1 / \alpha} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, \alpha > 0, the third shape parameter, and \sigma > 0, the scale parameter.

Usage

dkumweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)
pkumweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varkumweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
eskumweibull(p, a=1, b=1, alpha=1, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dkumweibull(x)
pkumweibull(x)
varkumweibull(x)
eskumweibull(x)

Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Laplace distribution due to due to Laplace (1774) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{2 \sigma} \exp \left( -\frac {\mid x - \mu \mid}{\sigma} \right), \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} \exp \left( \frac {x - \mu}{\sigma} \right), & \mbox{if $x < \mu$,} \\ \\ \displaystyle 1 - \frac {1}{2} \exp \left( -\frac {x - \mu}{\sigma} \right), & \mbox{if $x \geq \mu$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \mu + \sigma \log (2 p), & \mbox{if $p < 1/2$,} \\ \\ \displaystyle \mu - \sigma \log \left[ 2 (1 - p) \right], & \mbox{if $p \geq 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \mu + \sigma \left[ \log (2 p) - 1 \right], & \mbox{if $p < 1/2$,} \\ \\ \displaystyle \mu + \sigma - \frac {\sigma}{p} + \sigma \frac {1 - p}{p} \log (1 - p) +\sigma \frac {1 - p}{p} \log 2, & \mbox{if $p \geq 1/2$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter.

Usage

dlaplace(x, mu=0, sigma=1, log=FALSE)
plaplace(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varlaplace(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eslaplace(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlaplace(x)
plaplace(x)
varlaplace(x)
eslaplace(x)

Linear failure rate distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the linear failure rate distribution due to Bain (1974) given by

\begin{array}{ll} &\displaystyle f(x) = (a + b x) \exp \left( -a x - b x^2 / 2 \right), \\ &\displaystyle F (x) = 1 - \exp \left( -a x - b x^2 / 2 \right), \\ &\displaystyle {\rm VaR}_p (X) = \frac {-a + \sqrt{a^2 - 2 b \log (1 - p)}}{b}, \\ &\displaystyle {\rm ES}_p (X) = -\frac {a}{b} + \frac {1}{b p} \int_0^p \sqrt{a^2 - 2 b \log (1 - v)} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first scale parameter, and b > 0, the second scale parameter.

Usage

dlfr(x, a=1, b=1, log=FALSE)
plfr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varlfr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eslfr(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlfr(x)
plfr(x)
varlfr(x)
eslfr(x)

Log beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log beta distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {(\log d - \log c)^{1 - a - b}}{x B (a, b)} (\log x - \log c)^{a - 1} (\log d - \log x)^{b - 1}, \\ &\displaystyle F (x) = I_{\frac {\log x - \log c}{\log d - \log c}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = c \left( \frac {d}{c} \right)^{I_p^{-1} (a, b)}, \\ &\displaystyle {\rm ES}_p (X) = \frac {c}{p} \int_0^p \left( \frac {d}{c} \right)^{I_v^{-1} (a, b)} dv \end{array}

for 0 < c \leq x \leq d, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, c > 0, the first location parameter, and d > 0, the second location parameter.

Usage

dlogbeta(x, a=1, b=1, c=1, d=2, log=FALSE)
plogbeta(x, a=1, b=1, c=1, d=2, log.p=FALSE, lower.tail=TRUE)
varlogbeta(p, a=1, b=1, c=1, d=2, log.p=FALSE, lower.tail=TRUE)
eslogbeta(p, a=1, b=1, c=1, d=2)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlogbeta(x)
plogbeta(x)
varlogbeta(x)
eslogbeta(x)

Log Cauchy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log Cauchy distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{x \pi} \frac {\sigma}{(\log x - \mu)^2 + \sigma^2}, \\ &\displaystyle F (x) = \frac {1}{\pi} \arctan \left( \frac {\log x - \mu}{\sigma} \right), \\ &\displaystyle {\rm VaR}_p (X) = \exp \left[ \mu + \sigma \tan \left( \pi p \right) \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {\exp (\mu)}{p} \int_0^p \exp \left[ \sigma \tan \left( \pi v \right) \right] dv \end{array}

for x > 0, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter.

Usage

dlogcauchy(x, mu=0, sigma=1, log=FALSE)
plogcauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varlogcauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eslogcauchy(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlogcauchy(x)
plogcauchy(x)
varlogcauchy(x)

 eslogcauchy(x)

Log gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log gamma distribution due to Consul and Jain (1971) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a^r x^{a - 1} (-\log x)^{r - 1}}{\Gamma (r)}, \\ &\displaystyle F (x) = Q (r, -a \log x), \\ &\displaystyle {\rm VaR}_p (X) = \exp \left[ -\frac {1}{a} Q^{-1} (r, p) \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \exp \left[ -\frac {1}{a} Q^{-1} (r, v) \right] dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, and r > 0, the second shape parameter.

Usage

dloggamma(x, a=1, r=1, log=FALSE)
ploggamma(x, a=1, r=1, log.p=FALSE, lower.tail=TRUE)
varloggamma(p, a=1, r=1, log.p=FALSE, lower.tail=TRUE)
esloggamma(p, a=1, r=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dloggamma(x)
ploggamma(x)
varloggamma(x)
esloggamma(x)

Logistic exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic exponential distribution due to Lan and Leemis (2008) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\displaystyle a \lambda \exp (\lambda x) \left[ \exp (\lambda x) - 1 \right]^{a - 1}} {\displaystyle \left\{ 1 + \left[ \exp (\lambda x) - 1 \right]^a \right\}^2}, \\ &\displaystyle F (x) = \frac {\displaystyle \left[ \exp (\lambda x) - 1 \right]^a} {\displaystyle 1 + \left[ \exp (\lambda x) - 1 \right]^a}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{\lambda} \log \left[ 1 + \left( \frac {p}{1 - p} \right)^{1 / a} \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p \lambda} \int_0^p \log \left[ 1 + \left( \frac {v}{1 - v} \right)^{1 / a} \right] dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter and \lambda > 0, the scale parameter.

Usage

dlogisexp(x, lambda=1, a=1, log=FALSE)
plogisexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
varlogisexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
eslogisexp(p, lambda=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlogisexp(x)
plogisexp(x)
varlogisexp(x)
eslogisexp(x)

Logistic Rayleigh distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic Rayleigh distribution due to Lan and Leemis (2008) given by

\begin{array}{ll} &\displaystyle f(x) = a \lambda x \exp \left( \lambda x^2 / 2 \right) \left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^{a - 1} \left\{ 1 + \left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^a \right\}^{-2}, \\ &\displaystyle F(x) = \frac {\left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^a} {1 + \left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^a}, \\ &\displaystyle {\rm VaR}_p (X) = \sqrt{\frac {2}{\lambda}} \sqrt{\log \left[ 1 + \left( \frac {p}{1 - p} \right)^{1 / a} \right]}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sqrt{2}}{p \sqrt{\lambda}} \int_0^p \left\{ \log \left[ 1 + \left( \frac {v}{1 - v} \right)^{1 / a} \right] \right\}^{1/2} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter, and \lambda > 0, the scale parameter.

Usage

dlogisrayleigh(x, a=1, lambda=1, log=FALSE)
plogisrayleigh(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varlogisrayleigh(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
eslogisrayleigh(p, a=1, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlogisrayleigh(x)
plogisrayleigh(x)
varlogisrayleigh(x)
eslogisrayleigh(x)

Logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma} \exp \left( -\frac {x - \mu}{\sigma} \right) \left[ 1 + \exp \left( -\frac {x - \mu}{\sigma} \right) \right]^{-2}, \\ &\displaystyle F (x) = \frac {1}{1 + \exp \left( -\frac {x - \mu}{\sigma} \right)}, \\ &\displaystyle {\rm VaR}_p (X) = \mu + \sigma \log \left[ p (1 - p) \right], \\ &\displaystyle {\rm ES}_p (X) = \mu - 2 \sigma + \sigma \log p - \sigma \frac {1 - p}{p} \log (1 - p) \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter.

Usage

dlogistic(x, mu=0, sigma=1, log=FALSE)
plogistic(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varlogistic(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eslogistic(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlogistic(x)
plogistic(x)
varlogistic(x)
eslogistic(x)

Log Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log Laplace distribution given by

\begin{array}{ll} &\displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {a b x^{b - 1}}{\delta^b (a + b)}, & \mbox{if $x \leq \delta$,} \\ \\ \displaystyle \frac {a b \delta^a}{x^{a + 1} (a + b)}, & \mbox{if $x > \delta$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {a x^b}{\delta^b (a + b)}, & \mbox{if $x \leq \delta$,} \\ \\ \displaystyle 1 - \frac {b \delta^a}{x^a (a + b)}, & \mbox{if $x > \delta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \delta \left[ p \frac {a + b}{a} \right]^{1/b}, & \mbox{if $p \leq \frac {a}{a + b}$,} \\ \\ \displaystyle \delta \left[ (1 - p) \frac {a + b}{a} \right]^{-1/a}, & \mbox{if $p > \frac {a}{a + b}$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \frac {\delta b}{b + 1} \left[ p \frac {a + b}{a} \right]^{1/b}, & \mbox{if $p \leq \frac {a}{a + b}$,} \\ \\ \displaystyle \frac {a \delta}{p (1 + 1/b) (a + b)} + \frac {a^{1/a} b^{1 - 1/a} \delta}{p (a + b) (1 - 1/a)} \\ \displaystyle \quad -\frac {\delta (1 - p)}{p (1 - 1/a)} \left[ \frac {a}{(a + b) (1 - p)} \right]^{1/a}, & \mbox{if $p > \frac {a}{a + b}$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, \delta > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dloglaplace(x, a=1, b=1, delta=0, log=FALSE)
ploglaplace(x, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)
varloglaplace(p, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)
esloglaplace(p, a=1, b=1, delta=0)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dloglaplace(x)
ploglaplace(x)
varloglaplace(x)
esloglaplace(x)

Loglog distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Loglog distribution due to Pham (2002) given by

\begin{array}{ll} &\displaystyle f(x) = a \log (\lambda) x^{a - 1} \lambda^{x^a} \exp \left[ 1 - \lambda^{x^a} \right], \\ &\displaystyle F (x) = 1 - \exp \left[ 1 - \lambda^{x^a} \right], \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \frac {\log \left[ 1 - \log (1 - p) \right]}{\log \lambda} \right\}^{1/a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p (\log \lambda)^{1/a}} \int_0^p \left\{ \log \left[ 1 - \log (1 - v) \right] \right\}^{1/a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter, and \lambda > 1, the scale parameter.

Usage

dloglog(x, a=1, lambda=2, log=FALSE)
ploglog(x, a=1, lambda=2, log.p=FALSE, lower.tail=TRUE)
varloglog(p, a=1, lambda=2, log.p=FALSE, lower.tail=TRUE)
esloglog(p, a=1, lambda=2)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dloglog(x)
ploglog(x)
varloglog(x)
esloglog(x)

Log-logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log-logistic distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b a^b x^{b - 1}} {\left( a^b + x^b \right)^2}, \\ &\displaystyle F (x) = \frac {x^b}{a^b + x^b}, \\ &\displaystyle {\rm VaR}_p (X) = a \left( \frac {p}{1 - p} \right)^{1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {a}{p} B_p \left( 1 + \frac {1}{b}, 1 - \frac {1}{b} \right) \end{array}

for x > 0, 0 < p < 1, a > 0, the scale parameter, and b > 0, the shape parameter, where B_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt denotes the incomplete beta function.

Usage

dloglogis(x, a=1, b=1, log=FALSE)
ploglogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varloglogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esloglogis(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dloglogis(x)
ploglogis(x)
varloglogis(x)
esloglogis(x)

Lognormal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the lognormal distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma x} \phi \left( \frac {\log x - \mu}{\sigma} \right), \\ &\displaystyle F (x) = \Phi \left( \frac {\log x - \mu}{\sigma} \right), \\ &\displaystyle {\rm VaR}_p (X) = \exp \left[ \mu + \sigma \Phi^{-1} (p) \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {\exp (\mu)}{p} \int_0^p \exp \left[ \sigma \Phi^{-1} (v) \right] dv \end{array}

for x > 0, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter.

Usage

dlognorm(x, mu=0, sigma=1, log=FALSE)
plognorm(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varlognorm(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eslognorm(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlognorm(x)
plognorm(x)
varlognorm(x)
eslognorm(x)

Lomax distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Lomax distribution due to Lomax (1954) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a}{\lambda} \left( 1 + \frac {x}{\lambda} \right)^{-a - 1}, \\ &\displaystyle F (x) = 1 - \left( 1 + \frac {x}{\lambda} \right)^{-a}, \\ &\displaystyle {\rm VaR}_p (X) = \lambda \left[ (1 - p)^{-1 / a} - 1 \right], \\ &\displaystyle {\rm ES}_p (X) = -\lambda + \frac {\lambda - \lambda (1 - p)^{1 - 1 / a}}{p - p / a} \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter, and \lambda > 0, the scale parameter.

Usage

dlomax(x, a=1, lambda=1, log=FALSE)
plomax(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varlomax(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
eslomax(p, a=1, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dlomax(x)
plomax(x)
varlomax(x)
eslomax(x)

Marshall-Olkin exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin exponential distribution due to Marshall and Olkin (1997) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\displaystyle \lambda \exp (\lambda x)} {\displaystyle \left[ \exp (\lambda x) - 1 + a \right]^2}, \\ &\displaystyle F (x) = \frac {\displaystyle \exp (\lambda x) - 2 + a}{\displaystyle \exp (\lambda x) - 1 + a}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{\lambda} \log \frac {2 - a - (1 - a) p}{1 - p}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{\lambda} \log \left[ 2 - a - (1 - a) p \right] - \frac {2 - a}{\lambda (1 - a) p} \log \frac {2 - a - (1 - a) p}{2 - a} + \frac {1 - p}{\lambda p} \log (1 - p) \end{array}

for x > 0, 0 < p < 1, a > 0, the first scale parameter and \lambda > 0, the second scale parameter.

Usage

dmoexp(x, lambda=1, a=1, log=FALSE)
pmoexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
varmoexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
esmoexp(p, lambda=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dmoexp(x)
pmoexp(x)
varmoexp(x)
esmoexp(x)

Marshall-Olkin Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin Weibull distribution due to Marshall and Olkin (1997) given by

\begin{array}{ll} &\displaystyle f(x) = b \lambda^b x^{b - 1} \exp \left[ (\lambda x)^b \right] \left\{ \exp \left[ (\lambda x)^b \right] - 1 + a \right\}^{-2}, \\ &\displaystyle F(x) = \frac {\displaystyle \exp \left[ (\lambda x)^b \right] - 2 + a} {\displaystyle \exp \left[ (\lambda x)^b \right] - 1 + a}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{\lambda} \left[ \log \left( \frac {1}{1 - p} + 1 - a \right) \right]^{1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{\lambda p} \int_0^p \left[ \log \left( \frac {1}{1 - v} + 1 - a \right) \right]^{1 / b} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first scale parameter, b > 0, the shape parameter, and \lambda > 0, the second scale parameter.

Usage

dmoweibull(x, a=1, b=1, lambda=1, log=FALSE)
pmoweibull(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varmoweibull(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
esmoweibull(p, a=1, b=1, lambda=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dmoweibull(x)
pmoweibull(x)
varmoweibull(x)
esmoweibull(x)

Nakagami distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Nakagami distribution due to Nakagami (1960) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {2 m^m}{\Gamma (m) a^m} x^{2 m - 1} \exp \left( -\frac {m x^2}{a} \right), \\ &\displaystyle F (x) = 1 - Q \left( m, \frac {m x^2}{a} \right), \\ &\displaystyle {\rm VaR}_p (X) = \sqrt{\frac {a}{m}} \sqrt{Q^{-1} (m, 1 - p)}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sqrt{a}}{p \sqrt{m}} \int_0^p \sqrt{Q^{-1} (m, 1 - v)} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the scale parameter, and m > 0, the shape parameter.

Usage

dnakagami(x, m=1, a=1, log=FALSE)
pnakagami(x, m=1, a=1, log.p=FALSE, lower.tail=TRUE)
varnakagami(p, m=1, a=1, log.p=FALSE, lower.tail=TRUE)
esnakagami(p, m=1, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dnakagami(x)
pnakagami(x)
varnakagami(x)
esnakagami(x)

Normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the normal distribution due to de Moivre (1738) and Gauss (1809) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\sigma} \phi \left( \frac {x - \mu}{\sigma} \right), \\ &\displaystyle F (x) = \Phi \left( \frac {x - \mu}{\sigma} \right), \\ &\displaystyle {\rm VaR}_p (X) = \mu + \sigma \Phi^{-1} (p), \\ &\displaystyle {\rm ES}_p (X) = \mu + \frac {\sigma}{p} \int_0^p \Phi^{-1} (v) dv \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \mu < \infty, the location parameter, and \sigma > 0, the scale parameter, where \phi (\cdot) denotes the pdf of a standard normal random variable, and \Phi (\cdot) denotes the cdf of a standard normal random variable.

Usage

dnormal(x, mu=0, sigma=1, log=FALSE)
pnormal(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varnormal(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esnormal(p, mu=0, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dnormal(x)
pnormal(x)
varnormal(x)
esnormal(x)

Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Pareto distribution due to Pareto (1964) given by

\begin{array}{ll} &\displaystyle f (x) = c K^c x^{-c - 1}, \\ &\displaystyle F (x) = 1 - \left( \frac {K}{x} \right)^c, \\ &\displaystyle {\rm VaR}_p (X) = K (1 - p)^{-1 / c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {K c}{p (1 - c)} (1 - p)^{1 - 1 / c} - \frac {K c}{p (1 - c)} \end{array}

for x \geq K, 0 < p < 1, K > 0, the scale parameter, and c > 0, the shape parameter.

Usage

dpareto(x, K=1, c=1, log=FALSE)
ppareto(x, K=1, c=1, log.p=FALSE, lower.tail=TRUE)
varpareto(p, K=1, c=1, log.p=FALSE, lower.tail=TRUE)
espareto(p, K=1, c=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dpareto(x)
ppareto(x)
varpareto(x)
espareto(x)

Pareto positive stable distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Pareto positive stable distribution due to Sarabia and Prieto (2009) and Guillen et al. (2011) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {\nu \lambda}{x} \left[ \log \left( \frac {x}{\sigma} \right) \right]^{\nu - 1} \exp \left\{ -\lambda \left[ \log \left( \frac {x}{\sigma} \right) \right]^\nu \right\}, \\ &\displaystyle F (x) = 1 - \exp \left\{ -\lambda \left[ \log \left( \frac {x}{\sigma} \right) \right]^\nu \right\}, \\ &\displaystyle {\rm VaR}_p (X) = \sigma \exp \left\{ \left[ -\frac {1}{\lambda} \log (1 - p) \right]^{1/\nu} \right\}, \\ &\displaystyle {\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \exp \left\{ \left[ -\frac {1}{\lambda} \log (1 - v) \right]^{1/\nu} \right\} dv \end{array}

for x > 0, 0 < p < 1, \lambda > 0, the first scale parameter, \sigma > 0, the second scale parameter, and \nu > 0, the shape parameter.

Usage

dparetostable(x, lambda=1, nu=1, sigma=1, log=FALSE)
pparetostable(x, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varparetostable(p, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esparetostable(p, lambda=1, nu=1, sigma=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dparetostable(x)
pparetostable(x)
varparetostable(x)
esparetostable(x)

Perks distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Perks distribution due to Perks (1932) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {\displaystyle a \exp (b x) \left[ 1 + a \right]} {\displaystyle \left[ 1 + a \exp (b x) \right]^2}, \\ &\displaystyle F (x) = 1 - \frac {\displaystyle 1 + a} {\displaystyle 1 + a \exp (b x)}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} \log \frac {a + p}{a (1 - p)}, \\ &\displaystyle {\rm ES}_p (X) = -\left( 1 + \frac {a}{p} \right) \frac {\log a}{b} +\frac {(a + p) \log (a +p)}{b p} + \frac {(1 - p) \log (1 - p)}{b p} \end{array}

for x > 0, 0 < p < 1, a > 0, the first scale parameter and b > 0, the second scale parameter.

Usage

dperks(x, a=1, b=1, log=FALSE)
pperks(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varperks(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esperks(p, a=1, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dperks(x)
pperks(x)
varperks(x)
esperks(x)

Power function I distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the power function I distribution given by

\begin{array}{ll} &\displaystyle f (x) = a x^{a - 1}, \\ &\displaystyle F (x) = x^a, \\ &\displaystyle {\rm VaR}_p (X) = p^{1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {p^{1 / a}}{1 / a + 1} \end{array}

for 0 < x < 1, 0 < p < 1, and a > 0, the shape parameter.

Usage

dpower1(x, a=1, log=FALSE)
ppower1(x, a=1, log.p=FALSE, lower.tail=TRUE)
varpower1(p, a=1, log.p=FALSE, lower.tail=TRUE)
espower1(p, a=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dpower1(x)
ppower1(x)
varpower1(x)
espower1(x)

Power function II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the power function II distribution given by

\begin{array}{ll} &\displaystyle f (x) = b (1 - x)^{b - 1}, \\ &\displaystyle F (x) = 1 - (1 - x)^b, \\ &\displaystyle {\rm VaR}_p (X) = 1 - (1 - p)^{1 / b}, \\ &\displaystyle {\rm ES}_p (X) = 1 + \frac {b \left[ (1 - p)^{1 / b + 1} - 1 \right]}{p (b + 1)} \end{array}

for 0 < x < 1, 0 < p < 1, and b > 0, the shape parameter.

Usage

dpower2(x, b=1, log=FALSE)
ppower2(x, b=1, log.p=FALSE, lower.tail=TRUE)
varpower2(p, b=1, log.p=FALSE, lower.tail=TRUE)
espower2(p, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dpower2(x)
ppower2(x)
varpower2(x)
espower2(x)

Quadratic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the quadratic distribution given by

\begin{array}{ll} &\displaystyle f(x) = \alpha (x - \beta)^2, \\ &\displaystyle F(x) = \frac {\alpha}{3} \left[ (x - \beta)^3 + (\beta - a)^3 \right], \\ &\displaystyle {\rm VaR}_p (X) = \beta + \left[ \frac {3 p}{\alpha} - (\beta - a)^3 \right]^{1/3}, \\ &\displaystyle {\rm ES}_p (X) = \beta + \frac {\alpha}{4 p} \left\{ \left[ \frac {3 p}{\alpha} - (\beta - a)^3 \right]^{4/3} - (\beta - a)^4 \right\} \end{array}

for a \leq x \leq b, 0 < p < 1, -\infty < a < \infty , the first location parameter, and -\infty < a < b < \infty, the second location parameter, where \alpha = \frac {12}{(b - a)^3} and \beta = \frac {a + b}{2}.

Usage

dquad(x, a=0, b=1, log=FALSE)
pquad(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)
varquad(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)
esquad(p, a=0, b=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dquad(x)
pquad(x)
varquad(x)
esquad(x)

Reflected gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the reflected gamma distribution due to Borgi (1965) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{\displaystyle 2 \phi \Gamma \left( a \right)} \left | \frac {x - \theta}{\phi} \right |^{a - 1} \exp \left\{ -\left | \frac {x - \theta}{\phi} \right | \right\}, \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} Q \left( a, \frac {\theta - x}{\phi} \right), & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle 1 - \frac {1}{2} Q \left( a, \frac {x - \theta}{\phi} \right), & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \phi Q^{-1} \left( a, 2 p \right), & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta + \phi Q^{-1} \left( a, 2 (1 - p) \right), & \mbox{if $p > 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta - \frac {\phi}{p} \int_0^p Q^{-1} \left( a, 2 v \right) dv, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta - \frac {\phi}{p} \int_0^{1/2} Q^{-1} \left( a, 2 v \right) dv +\frac {\phi}{p} \int_{1/2}^p Q^{-1} \left( a, 2 (1 - v) \right) dv, & \mbox{if $p > 1/2$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, \phi > 0, the scale parameter, and a > 0, the shape parameter.

Usage

drgamma(x, a=1, theta=0, phi=1, log=FALSE)
prgamma(x, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
varrgamma(p, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
esrgamma(p, a=1, theta=0, phi=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
drgamma(x)
prgamma(x)
varrgamma(x)
esrgamma(x)

Schabe distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Schabe distribution due to Schabe (1994) given by

\begin{array}{ll} &\displaystyle f(x) = \frac {\displaystyle 2 \gamma + (1 - \gamma) x / \theta}{\displaystyle \theta (\gamma + x/\theta)^2}, \\ &\displaystyle F(x) = \frac {\displaystyle (1 + \gamma) x}{\displaystyle x + \gamma \theta}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {p \gamma \theta}{1 + \gamma - p}, \\ &\displaystyle {\rm ES}_p (X) = -\theta \gamma - \frac {\theta \gamma (1 + \gamma)}{p} \log \frac {1 + \gamma - p}{1 + \gamma} \end{array}

for x > 0, 0 < p < 1, 0 < \gamma < 1, the first scale parameter, and \theta > 0, the second scale parameter.

Usage

dschabe(x, gamma=0.5, theta=1, log=FALSE)
pschabe(x, gamma=0.5, theta=1, log.p=FALSE, lower.tail=TRUE)
varschabe(p, gamma=0.5, theta=1, log.p=FALSE, lower.tail=TRUE)
esschabe(p, gamma=0.5, theta=1)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dschabe(x)
pschabe(x)
varschabe(x)
esschabe(x)

Hyperbolic secant distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the hyperbolic secant distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{2} {\rm sech} \left( \frac {\pi x}{2} \right), \\ &\displaystyle F (x) = \frac {2}{\pi} \arctan \left[ \exp \left( \frac {\pi x}{2} \right) \right], \\ &\displaystyle {\rm VaR}_p (X) = \frac {2}{\pi} \log \left[ \tan \left( \frac {\pi p}{2} \right) \right], \\ &\displaystyle {\rm ES}_p (X) = \frac {2}{\pi p} \int_0^p \log \left[ \tan \left( \frac {\pi v}{2} \right) \right] dv \end{array}

for -\infty < x < \infty, and 0 < p < 1.

Usage

dsecant(x, log=FALSE)
psecant(x, log.p=FALSE, lower.tail=TRUE)
varsecant(p, log.p=FALSE, lower.tail=TRUE)
essecant(p)

Arguments

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658