Distributacalcul: Probability Distribution Functions

Description

The Distributacalcul provides three general types of functions :

1. Functions related to continuous, discrete, and compound probability distributions.

2. Functions related to bivariate copulas.

3. Shiny applications for visualising these distributions and copulas.

Probability Distribution Functions

These include functions for calculating various moments, risk measures, and distribution functions.

Copula Functions

These include functions for calculating distribution functions and simulating observations from the copulas.

Author(s)

Maintainer: Alec James van Rassel alec.van-rassel.1@ulaval.ca [copyright holder]

Authors:

• Gabriel Crépeault-Cauchon [conceptor]

Other contributors:

• Étienne Marceau [teacher, scientific advisor]

• Hélène Cossette [teacher, scientific advisor]

• Laboratoire Act & Risk [funder, supporting host]

• École d'actuariat de l'Université Laval [funder, host institution, university place]

• Natural Sciences and Engineering Research Council of Canada [funder]

• Marc-André Devost [conceptor]

See Also

Useful links:

• https://alec42.github.io/Distributacalcul_Package/

• Report bugs at https://github.com/alec42/Distributacalcul_Package/issues

Beta Distribution

Description

Beta distribution with shape parameters \alpha and \beta.

Usage

expValBeta(shape1, shape2)

varBeta(shape1, shape2)

kthMomentBeta(k, shape1, shape2)

expValLimBeta(d, shape1, shape2)

expValTruncBeta(d, shape1, shape2, less.than.d = TRUE)

stopLossBeta(d, shape1, shape2)

meanExcessBeta(d, shape1, shape2)

VatRBeta(kap, shape1, shape2)

TVatRBeta(kap, shape1, shape2)

mgfBeta(t, shape1, shape2, k0)

Arguments

Details

The Beta distribution with shape parameters \alpha and \beta has density:

f\left(x\right) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) % \Gamma(\beta)} x^{\alpha - 1} (1 - x)^(\beta - 1)

for x \in [0, 1], \alpha, \beta > 0.

Value

Function :

• expValBeta gives the expected value.

• varBeta gives the variance.

• kthMomentBeta gives the kth moment.

• expValLimBeta gives the limited mean.

• expValTruncBeta gives the truncated mean.

• stopLossBeta gives the stop-loss.

• meanExcessBeta gives the mean excess loss.

• VatRBeta gives the Value-at-Risk.

• TVatRBeta gives the Tail Value-at-Risk.

• mgfBeta gives the moment generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRBeta is a wrapper for the qbeta function from the stats package.

Examples

expValBeta(shape1 = 3, shape2 = 5)

varBeta(shape1 = 4, shape2 = 5)

kthMomentBeta(k = 3, shape1 = 4, shape2 = 5)

expValLimBeta(d = 0.3, shape1 = 4, shape2 = 5)

expValTruncBeta(d = 0.4, shape1 = 4, shape2 = 5)

# Values less than d
expValTruncBeta(d = 0.4, shape1 = 4, shape2 = 5, less.than.d = FALSE)

stopLossBeta(d = 0.3, shape1 = 4, shape2 = 5)

meanExcessBeta(d = .3, shape1 = 4, shape2 = 5)

VatRBeta(kap = .99, shape1 = 4, shape2 = 5)

TVatRBeta(kap = .99, shape1 = 4, shape2 = 5)

mgfBeta(t = 1, shape1 = 3, shape2 = 5, k0 = 1E2)

Compound Binomial Distribution

Description

Computes various risk measures (mean, variance, Value-at-Risk (VaR), and Tail Value-at-Risk (TVaR)) for the compound Binomial distribution.

Usage

pCompBinom(
  x,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

expValCompBinom(
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

varCompBinom(
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

VatRCompBinom(
  kap,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

TVatRCompBinom(
  kap,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  vark,
  k0,
  distr_severity = "Gamma"
)

Arguments

Details

The compound binomial distribution has density ....

Value

Function :

• pCompBinom gives the cumulative density function.

• expValCompBinom gives the expected value.

• varCompBinom gives the variance.

• TVatRCompBinom gives the Tail Value-at-Risk.

• VatRCompBinom gives the Value-at-Risk.

Returned values are approximations for the cumulative density function, TVaR, and VaR.

Examples

pCompBinom(x = 2, size = 1, prob = 0.2, shape = log(1000) - 0.405,
          rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")

expValCompBinom(size = 1, prob = 0.2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")

varCompBinom(size = 1, prob = 0.2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")

VatRCompBinom(kap = 0.9, size = 1, prob = 0.2, shape = log(1000) - 0.405,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")

vark_calc <- VatRCompBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
TVatRCompBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59, rate = 0.9^2,
            vark = vark_calc, k0 = 1E2, distr_severity = "Gamma")

Compound Negative Binomial Distribution

Description

Computes various risk measures (mean, variance, Value-at-Risk (VatR), and Tail Value-at-Risk (TVatR)) for the compound Negative Binomial distribution.

Usage

pCompNBinom(
  x,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

expValCompNBinom(
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

varCompNBinom(
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

VatRCompNBinom(
  kap,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

TVatRCompNBinom(
  kap,
  vark,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

Arguments

Details

The compound negative binomial distribution has density ....

Value

Function :

• pCompNBinom gives the cumulative density function.

• expValCompNBinom gives the expected value.

• varCompNBinom gives the variance.

• TVatRCompNBinom gives the Tail Value-at-Risk.

• VatRCompNBinom gives the Value-at-Risk.

Returned values are approximations for the cumulative density function, TVatR, and VatR.

Examples

pCompNBinom(x = 2, size = 1, prob = 0.2, shape = log(1000) - 0.405,
          rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")


expValCompNBinom(size = 4, prob = 0.2, shape = 0, scale = 1,
         distr_severity = "Lognormal")

varCompNBinom(size = 1, prob = 0.2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")

VatRCompNBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")

vark_calc <- VatRCompNBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
TVatRCompNBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59, rate = 0.9^2,
            vark = vark_calc, k0 = 1E2, distr_severity = "Gamma")

Compound Poisson Distribution

Description

Computes various risk measures (mean, variance, Value-at-Risk (VaR), and Tail Value-at-Risk (TVaR)) for the compound Poisson distribution.

Usage

pCompPois(
  x,
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

expValCompPois(
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

varCompPois(
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

VatRCompPois(
  kap,
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

TVatRCompPois(
  kap,
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  vark,
  k0,
  distr_severity = "Gamma"
)

Arguments

Details

The compound Poisson distribution with parameters ... has density ....

Value

Function :

• pCompPois gives the cumulative density function.

• expValCompPois gives the expected value.

• varCompPois gives the variance.

• TVatRCompPois gives the Tail Value-at-Risk.

• VatRCompPois gives the Value-at-Risk.

Returned values are approximations for the cumulative density function, TVaR, and VaR.

Examples

pCompPois(x = 2, lambda = 2, shape = log(1000) - 0.405,
          rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")

expValCompPois(lambda = 2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")

varCompPois(lambda = 2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")

VatRCompPois(kap = 0.9, lambda = 2, shape = log(1000) - 0.405,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")

vark_calc <- VatRCompPois(kap = 0.9, lambda = 2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
TVatRCompPois(kap = 0.9, lambda = 2, shape = 0.59, rate = 0.9^2,
            vark = vark_calc, k0 = 1E2, distr_severity = "Gamma")

Hypergeometric Distribution

Description

Hypergeometric distribution where we have a sample of k balls from an urn containing N, of which m are white and n are black.

Usage

expValErl(N = n + m, m, n = N - m, k)

varErl(N = n + m, m, n = N - m, k)

Arguments

Details

The Hypergeometric distribution for N total items of which m are of one type and n of the other and from which k items are picked has probability mass function :

Pr(X = x) = \frac{\left(\frac{m}{k}\right)\left(\frac{n}{k - x}\right)}{\left(\frac{N}{k}\right)}

for x = 0, 1, \dots, \min(k, m).

Value

Function :

• expValErl gives the expected value.

• varErl gives the variance.

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

# With total balls specified
expValErl(N = 5, m = 2, k = 2)

# With number of each colour of balls specified
expValErl(m = 2, n = 3, k = 2)


# With total balls specified
varErl(N = 5, m = 2, k = 2)

# With number of each colour of balls specified
varErl(m = 2, n = 3, k = 2)

Exponential Distribution

Description

Exponential distribution with rate parameter \beta.

Usage

expValExp(rate = 1/scale, scale = 1/rate)

varExp(rate = 1/scale, scale = 1/rate)

kthMomentExp(k, rate = 1/scale, scale = 1/rate)

expValLimExp(d, rate = 1/scale, scale = 1/rate)

expValTruncExp(d, rate = 1/scale, scale = 1/rate, less.than.d = TRUE)

stopLossExp(d, rate = 1/scale, scale = 1/rate)

meanExcessExp(d, rate = 1/scale, scale = 1/rate)

VatRExp(kap, rate = 1/scale, scale = 1/rate)

TVatRExp(kap, rate = 1/scale, scale = 1/rate)

mgfExp(t, rate = 1/scale, scale = 1/rate)

Arguments

Details

The Exponential distribution with rate parameter \beta has density:

f\left(x\right) = \frac{1}{\beta}\textrm{e}^{-\beta x}

for x \in \mathcal{R}^+, \beta > 0.

Value

Function :

• expValExp gives the expected value.

• varExp gives the variance.

• kthMomentExp gives the kth moment.

• expValLimExp gives the limited mean.

• expValTruncExp gives the truncated mean.

• stopLossExp gives the stop-loss.

• meanExcessExp gives the mean excess loss.

• VatRExp gives the Value-at-Risk.

• TVatRExp gives the Tail Value-at-Risk.

• mgfExp gives the moment generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRExp is a wrapper of the qexp function from the stats package.

Examples

# With scale parameter
expValExp(scale = 4)

# With rate parameter
expValExp(rate = 0.25)

# With scale parameter
varExp(scale = 4)

# With rate parameter
varExp(rate = 0.25)

# With scale parameter
kthMomentExp(k = 2, scale = 4)

# With rate parameter
kthMomentExp(k = 2, rate = 0.25)

# With scale parameter
expValLimExp(d = 2, scale = 4)

# With rate parameter
expValLimExp(d = 2, rate = 0.25)

# With scale parameter
expValTruncExp(d = 2, scale = 4)

# With rate parameter, values greater than d
expValTruncExp(d = 2, rate = 0.25, less.than.d = FALSE)

# With scale parameter
stopLossExp(d = 2, scale = 4)

# With rate parameter
stopLossExp(d = 2, rate = 0.25)

# With scale parameter
meanExcessExp(d = 2, scale = 4)

# With rate parameter
meanExcessExp(d = 5, rate = 0.25)

# With scale parameter
VatRExp(kap = .99, scale = 4)

# With rate parameter
VatRExp(kap = .99, rate = 0.25)

# With scale parameter
TVatRExp(kap = .99, scale = 4)

# With rate parameter
TVatRExp(kap = .99, rate = 0.25)

mgfExp(t = 1, rate = 5)

Gamma Distribution

Description

Gamma distribution with shape parameter \alpha and rate parameter \beta.

Usage

expValGamma(shape, rate = 1/scale, scale = 1/rate)

varGamma(shape, rate = 1/scale, scale = 1/rate)

kthMomentGamma(k, shape, rate = 1/scale, scale = 1/rate)

expValLimGamma(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncGamma(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE)

stopLossGamma(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessGamma(d, shape, rate = 1/scale, scale = 1/rate)

VatRGamma(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRGamma(kap, shape, rate = 1/scale, scale = 1/rate)

mgfGamma(t, shape, rate = 1/scale, scale = 1/rate)

Arguments

Details

The Gamma distribution with shape parameter \alpha and rate parameter \beta has density:

f\left(x\right) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1}% \textrm{e}^{-\beta x}

for x \in \mathcal{R}^+, \beta, \alpha > 0.

Value

Function :

• expValGamma gives the expected value.

• varGamma gives the variance.

• kthMomentGamma gives the kth moment.

• expValLimGamma gives the limited mean.

• expValTruncGamma gives the truncated mean.

• stopLossGamma gives the stop-loss.

• meanExcessGamma gives the mean excess loss.

• VatRGamma gives the Value-at-Risk.

• TVatRGamma gives the Tail Value-at-Risk.

• mgfGamma gives the moment generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRGamma is a wrapper for the qgamma function stats package.

Examples

# With scale parameter
expValGamma(shape = 3, scale = 4)

# With rate parameter
expValGamma(shape = 3, rate = 0.25)


# With scale parameter
varGamma(shape = 3, scale = 4)

# With rate parameter
varGamma(shape = 3, rate = 0.25)


# With scale parameter
kthMomentGamma(k = 2, shape = 3, scale = 4)

# With rate parameter
kthMomentGamma(k = 2, shape = 3, rate = 0.25)


# With scale parameter
expValLimGamma(d = 2, shape = 3, scale = 4)

# With rate parameter
expValLimGamma(d = 2, shape = 3, rate = 0.25)


# With scale parameter
expValTruncGamma(d = 2, shape = 3, scale = 4)

# With rate parameter
expValTruncGamma(d = 2, shape = 3, rate = 0.25)

# values greather than d
expValTruncGamma(d = 2, shape = 3, rate = 0.25, less.than.d = FALSE)


# With scale parameter
stopLossGamma(d = 2, shape = 3, scale = 4)

# With rate parameter
stopLossGamma(d = 2, shape = 3, rate = 0.25)


# With scale parameter
meanExcessGamma(d = 2, shape = 3, scale = 4)

# With rate parameter
meanExcessGamma(d = 2, shape = 3, rate = 0.25)


# With scale parameter
VatRGamma(kap = .2, shape = 3, scale = 4)

# With rate parameter
VatRGamma(kap = .2, shape = 3, rate = 0.25)


# With scale parameter
TVatRGamma(kap = .2, shape = 3, scale = 4)

# With rate parameter
TVatRGamma(kap = .2, shape = 3, rate = 0.25)


mgfGamma(t = 1, shape = 3, rate = 5)

Inverse Gaussian Distribution

Description

Inverse Gaussian distribution with mean \mu and shape parameter \beta.

Usage

expValIG(mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

varIG(mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

expValLimIG(d, mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

expValTruncIG(
  d,
  mean,
  shape = dispersion * mean^2,
  dispersion = shape/mean^2,
  less.than.d = TRUE
)

stopLossIG(d, mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

meanExcessIG(d, mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

VatRIG(kap, mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

TVatRIG(kap, mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

mgfIG(t, mean, shape = dispersion * mean^2, dispersion = shape/mean^2)

Arguments

Details

The Inverse Gaussian distribution with

Value

Function :

• expValIG gives the expected value.

• varIG gives the variance.

• expValLimIG gives the limited mean.

• expValTruncIG gives the truncated mean.

• stopLossIG gives the stop-loss.

• meanExcessIG gives the mean excess loss.

• VatRIG gives the Value-at-Risk.

• TVatRIG gives the Tail Value-at-Risk.

• mgfIG gives the moment generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRIG is a wrapper for the qinvgauss function from the statmod package.

Examples

expValIG(mean = 2, shape = 5)

varIG(mean = 2, shape = 5)

expValLimIG(d = 2, mean = 2, shape = 5)

expValTruncIG(d = 2, mean = 2, shape = 5)

stopLossIG(d = 2, mean = 2, shape = 5)

meanExcessIG(d = 2, mean = 2, shape = 5)

VatRIG(kap = 0.99, mean = 2, shape = 5)

TVatRIG(kap = 0.99, mean = 2, shape = 5)

mgfIG(t = 1, mean = 2, shape = .5)

Lognormal Distribution

Description

Lognormal distribution with mean \mu and variance \sigma.

Usage

expValLnorm(meanlog, sdlog)

varLnorm(meanlog, sdlog)

kthMomentLnorm(k, meanlog, sdlog)

expValLimLnorm(d, meanlog, sdlog)

expValTruncLnorm(d, meanlog, sdlog, less.than.d = TRUE)

stopLossLnorm(d, meanlog, sdlog)

meanExcessLnorm(d, meanlog, sdlog)

VatRLnorm(kap, meanlog, sdlog)

TVatRLnorm(kap, meanlog, sdlog)

Arguments

Details

The Log-normal distribution with mean \mu and standard deviation \sigma has density:

\frac{1}{\sqrt{2\pi}\sigma x}\textrm{e}^{-\frac{1}{2}\left(\frac{\ln(x) - \mu}{\sigma}\right)^2}

for x \in \mathcal{R}^{+}, \mu \in \mathcal{R}, \sigma > 0.

Value

Function :

• expValLnorm gives the expected value.

• varLnorm gives the variance.

• kthMomentLnorm gives the kth moment.

• expValLimLnorm gives the limited mean.

• expValTruncLnorm gives the truncated mean.

• stopLossLnorm gives the stop-loss.

• meanExcessLnorm gives the mean excess loss.

• VatRLnorm gives the Value-at-Risk.

• TVatRLnorm gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRLnorm is a wrapper of the qlnorm function from the stats package.

Examples

expValLnorm(meanlog = 3, sdlog = 5)

varLnorm(meanlog = 3, sdlog = 5)

kthMomentLnorm(k = 2, meanlog = 3, sdlog = 5)

expValLimLnorm(d = 2, meanlog = 2, sdlog = 5)

expValTruncLnorm(d = 2, meanlog = 2, sdlog = 5)

# Values greater than d
expValTruncLnorm(d = 2, meanlog = 2, sdlog = 5, less.than.d = FALSE)

stopLossLnorm(d = 2, meanlog = 2, sdlog = 5)

meanExcessLnorm(d = 2, meanlog = 2, sdlog = 5)

VatRLnorm(kap = 0.8, meanlog = 3, sdlog = 5)

TVatRLnorm(kap = 0.8, meanlog = 2, sdlog = 5)

Logarithmic Distribution

Description

Logarithmic distribution with probability parameter \gamma.

Usage

dLogarithmic(x, prob)

pLogarithmic(q, prob, lower.tail = TRUE)

expValLogarithmic(prob)

varLogarithmic(prob)

VatRLogarithmic(kap, prob)

mgfLogarithmic(t, prob)

pgfLogarithmic(t, prob)

Arguments

Details

The Logarithmic distribution with probability parameter \gamma has probability mass function :

Pr(X = k) = \frac{-\gamma^{k}}{\ln(1 - \gamma)k}

, for k = 0, 1, 2, \dots, and \gamma \in (0, 1)].

Value

Function :

• dLogarithmic gives the probability density function (PDF).

• pLogarithmic gives the cumulative density function (CDF).

• expValLogarithmic gives the expected value.

• varLogarithmic gives the variance.

• VatRLogarithmic gives the Value-at-Risk.

• mgfLogarithmic gives the moment generating function (MGF).

• pgfLogarithmic gives the probability generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

dLogarithmic(x = 3, prob = 0.2)

pLogarithmic(q = 3, prob = 0.2)

expValLogarithmic(prob = 0.50)

varLogarithmic(prob = 0.50)

VatRLogarithmic(kap = 0.99, prob = 0.2)

mgfLogarithmic(t = .2, prob = 0.50)

pgfLogarithmic(t = .2, prob = 0.50)

Negative Binomial Distribution

Description

Negative binomial distribution with parameters r (number of successful trials) and p (probability of success).

Usage

expValNBinom(
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

varNBinom(
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

mgfNBinom(
  t,
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

pgfNBinom(
  t,
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

Arguments

Details

When k is the number of failures until the rth success, with a probability p of a success, the negative binomial has density:

\left(\frac{r + k - 1}{k}\right) (p)^{r} (1 - p)^{k}

for k \in \{0, 1, \dots \}

When k is the number of trials until the rth success, with a probability p of a success, the negative binomial has density:

\left(\frac{k - 1}{r - 1}\right) (p)^{r} (1 - p)^{k - r}

for k \in \{r, r + 1, r + 2, \dots \}

The alternative parameterization of the negative binomial with parameter \beta, and k being the number of trials, has density:

\frac{\Gamma(r + k)}{\Gamma(r) k!} \left(\frac{1}{1 + \beta}\right)^{r}% \left(\frac{\beta}{1 + \beta}\right)^{k - r}

for k \in \{0, 1, \dots \}

Value

Function :

• expValNBinom gives the expected value.

• varNBinom gives the variance.

• mgfNBinom gives the moment generating function (MGF).

• pgfNBinom gives the probability generating function (PGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

# Where k is the number of trials for a rth success
expValNBinom(size = 2, prob = .4)

# Where k is the number of failures before a rth success
expValNBinom(size = 2, prob = .4, nb_tries = TRUE)

# With alternative parameterization where k is the number of trials
expValNBinom(size = 2, beta = 1.5)


# Where k is the number of trials for a rth success
varNBinom(size = 2, prob = .4)

# Where k is the number of failures before a rth success
varNBinom(size = 2, prob = .4, nb_tries = TRUE)

# With alternative parameterization where k is the number of trials
varNBinom(size = 2, beta = 1.5)

mgfNBinom(t = 1, size = 4, prob = 0.5)

pgfNBinom(t = 5, size = 3, prob = 0.3)

Normal Distribution

Description

Normal distribution

Usage

expValNorm(mean, sd)

varNorm(mean, sd)

expValLimNorm(d, mean = 0, sd = 1)

expValTruncNorm(d, mean = 0, sd = 1, less.than.d = TRUE)

stopLossNorm(d, mean = 0, sd = 1)

meanExcessNorm(d, mean = 0, sd = 1)

VatRNorm(kap, mean = 0, sd = 1)

TVatRNorm(kap, mean = 0, sd = 1)

mgfNorm(t, mean = 0, sd = 1)

Arguments

Details

The Normal distribution with mean \mu and standard deviation \sigma has density:

\frac{1}{\sqrt{2\pi}\sigma}\textrm{e}^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}

for x \in \mathcal{R}, \mu \in \mathcal{R}, \sigma > 0.

Value

Function :

• expValNorm gives the expected value.

• varNorm gives the variance.

• expValLimNorm gives the limited mean.

• expValTruncNorm gives the truncated mean.

• stopLossNorm gives the stop-loss.

• meanExcessNorm gives the mean excess loss.

• VatRNorm gives the Value-at-Risk.

• TVatRNorm gives the Tail Value-at-Risk.

• mgfNorm gives the moment generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRNorm is a wrapper of the qnorm function from the stats package.

Examples

expValNorm(mean = 3, sd = 5)

varNorm(mean = 3, sd = 5)

expValLimNorm(d = 2, mean = 2, sd = 5)

expValTruncNorm(d = 2, mean = 2, sd = 5)

stopLossNorm(d = 2, mean = 2, sd = 5)

meanExcessNorm(d = 2, mean = 2, sd = 5)

VatRNorm(kap = 0.8, mean = 3, sd = 5)

TVatRNorm(kap = 0.8, mean = 2, sd = 5)

mgfNorm(t = 1, mean = 3, sd = 5)

Pareto Distribution

Description

Pareto distribution with shape parameter \alpha and rate parameter \lambda.

Usage

dPareto(x, shape, rate = 1/scale, scale = 1/rate)

pPareto(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE)

expValPareto(shape, rate = 1/scale, scale = 1/rate)

varPareto(shape, rate = 1/scale, scale = 1/rate)

kthMomentPareto(k, shape, rate = 1/scale, scale = 1/rate)

expValLimPareto(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncPareto(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE)

stopLossPareto(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessPareto(d, shape, rate = 1/scale, scale = 1/rate)

VatRPareto(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRPareto(kap, shape, rate = 1/scale, scale = 1/rate)

Arguments

Details

The Pareto distribution with rate parameter \lambda as well as shape parameter \alpha has density:

f\left(x\right) = \frac{\alpha\lambda^{\alpha}}% {(\lambda + x)^{\alpha + 1}}

for x \in \mathcal{R}^+, \alpha, \lambda > 0.

Value

Function :

• dPareto gives the probability density function (PDF).

• pPareto gives the cumulative density function (CDF).

• expValPareto gives the expected value.

• varPareto gives the variance.

• kthMomentPareto gives the kth moment.

• expValLimPareto gives the limited mean.

• expValTruncPareto gives the truncated mean.

• stopLossPareto gives the stop-loss.

• meanExcessPareto gives the mean excess loss.

• VatRPareto gives the Value-at-Risk.

• TVatRPareto gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

# With scale parameter
dPareto(x = 2, shape = 2, scale = 5)

# With rate parameter
dPareto(x = 2, shape = 2, rate = .20)

# With scale parameter
pPareto(q = 2, shape = 2, scale = 5)

# With rate parameter
pPareto(q = 2, shape = 2, rate = 0.20)

# Survival function
pPareto(q = 2, shape = 2, rate = 0.20, lower.tail = FALSE)

# With scale parameter
expValPareto(shape = 5, scale = 0.5)

# With rate parameter
expValPareto(shape = 5, rate = 2)

# With scale parameter
varPareto(shape = 5, scale = 0.5)

# With rate parameter
varPareto(shape = 5, rate = 2)

# With scale parameter
kthMomentPareto(k = 4, shape = 5, scale = 0.5)

# With rate parameter
kthMomentPareto(k = 4, shape = 5, rate = 2)

# With scale parameter
expValLimPareto(d = 4, shape = 5, scale = 0.5)

# With rate parameter
expValLimPareto(d = 4, shape = 5, rate = 2)

# With scale parameter
expValTruncPareto(d = 4, shape = 5, scale = 0.5)

# With rate parameter
expValTruncPareto(d = 4, shape = 5, rate = 2)

# With scale parameter
stopLossPareto(d = 2, shape = 5, scale = 0.5)

# With rate parameter
stopLossPareto(d = 2, shape = 5, rate = 2)

# With scale parameter
meanExcessPareto(d = 6, shape = 5, scale = 0.5)

# With rate parameter
meanExcessPareto(d = 6, shape = 5, rate = 2)

# With scale parameter
VatRPareto(kap = .99, shape = 5, scale = 0.5)

# With rate parameter
VatRPareto(kap = .99, shape = 5, rate = 2)

# With scale parameter
TVatRPareto(kap = .99, shape = 5, scale = 0.5)

# With rate parameter
TVatRPareto(kap = .99, shape = 5, rate = 2)

Poisson Distribution

Description

Poisson distribution with rate parameter \lambda.

Usage

expValPois(lambda)

varPois(lambda)

expValTruncPois(d, lambda, k0, less.than.d = TRUE)

TVatRPois(kap, lambda, k0)

mgfPois(t, lambda)

pgfPois(t, lambda)

Arguments

Details

The Poisson distribution with rate parameter \lambda has probability mass function :

Pr(X = k) = \frac{\lambda^k \textrm{e}^{-\lambda}}{k!}

for k = 0, 1, 2, \dots, and \lambda > 0

Value

Function :

• expValPois gives the expected value.

• varPois gives the variance.

• expValTruncPois gives the truncated mean.

• TVatRPois gives the Tail Value-at-Risk.

• mgfPois gives the moment generating function (MGF).

• pgfPois gives the probability generating function (PGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

expValPois(lambda = 3)

varPois(lambda = 3)

expValTruncPois(d = 0, lambda = 2, k0 = 2E2, less.than.d = FALSE)
expValTruncPois(d = 2, lambda = 2, k0 = 2E2, less.than.d = TRUE)

TVatRPois(kap = 0.8, lambda = 3, k0 = 2E2)

mgfPois(t = 1, lambda = 3)

pgfPois(t = 1, lambda = 3)

Uniform Distribution

Description

Uniform distribution with min a and max b.

Usage

expValUnif(min = 0, max = 1)

varUnif(min = 0, max = 1)

kthMomentUnif(k, min = 0, max = 1)

expValLimUnif(d, min = 0, max = 1)

expValTruncUnif(d, min = 0, max = 1, less.than.d = TRUE)

stopLossUnif(d, min = 0, max = 1)

meanExcessUnif(d, min = 0, max = 1)

VatRUnif(kap, min = 0, max = 1)

TVatRUnif(kap, min = 0, max = 1)

mgfUnif(t, min = 0, max = 1)

Arguments

Details

The (continuous) uniform distribution with min and max parameters a and b respectively has density:

f(x) = \frac{1}{b - a} \times \bm{1}_{\{x \in [a, b] \}}

for x \in [a, b].

Value

Function :

• expValUnif gives the expected value.

• varUnif gives the variance.

• kthMomentUnif gives the kth moment.

• expValLimUnif gives the limited mean.

• expValTruncUnif gives the truncated mean.

• stopLossUnif gives the stop-loss.

• meanExcessUnif gives the mean excess loss.

• VatRUnif gives the Value-at-Risk.

• TVatRUnif gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

expValUnif(min = 3, max = 4)

varUnif(min = 3, max = 4)

kthMomentUnif(k = 2, min = 3, max = 4)

expValLimUnif(d = 3, min = 2, max = 4)

expValTruncUnif(d = 3, min = 2, max = 4)

# Values greather than d
expValTruncUnif(d = 3, min = 2, max = 4, less.than.d = FALSE)

stopLossUnif(d = 3, min = 2, max = 4)

meanExcessUnif(d = 2, min = 2, max = 4)

VatRUnif(kap = .99, min = 3, max = 4)

TVatRUnif(kap = .99, min = 3, max = 4)

mgfUnif(t = 2, min = 0, max = 1)

Weibull Distribution

Description

Weibull distribution with shape parameter \tau and rate parameter \beta.

Usage

expValWeibull(shape, rate = 1/scale, scale = 1/rate)

varWeibull(shape, rate = 1/scale, scale = 1/rate)

kthMomentWeibull(k, shape, rate = 1/scale, scale = 1/rate)

expValLimWeibull(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncWeibull(
  d,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  less.than.d = TRUE
)

stopLossWeibull(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessWeibull(d, shape, rate = 1/scale, scale = 1/rate)

VatRWeibull(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRWeibull(kap, shape, rate = 1/scale, scale = 1/rate)

Arguments

Details

The Weibull distribution with shape parameter \tau and rate parameter \beta has density:

f\left(x\right) = \beta \tau \left( \beta x \right) ^{\tau -1} % \mathrm{e}^{-\left( \beta x\right) ^{\tau }}

for x \in \mathcal{R}^+, \beta > 0, \tau > 0

Value

Function :

• expValWeibull gives the expected value.

• varWeibull gives the variance.

• kthMomentWeibull gives the kth moment.

• expValLimWeibull gives the limited mean.

• expValTruncWeibull gives the truncated mean.

• stopLossWeibull gives the stop-loss.

• meanExcessWeibull gives the mean excess loss.

• VatRWeibull gives the Value-at-Risk.

• TVatRWeibull gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

# With scale parameter
expValWeibull(shape = 2, scale = 5)

# With rate parameter
expValWeibull(shape = 2, rate = 0.2)

# With scale parameter
varWeibull(shape = 2, scale = 5)

# With rate parameter
varWeibull(shape = 2, rate = 0.2)

# With scale parameter
kthMomentWeibull(k = 2, shape = 2, scale = 5)

# With rate parameter
kthMomentWeibull(k = 2, shape = 2, rate = 0.2)

# With scale parameter
expValLimWeibull(d = 2, shape = 2, scale = 5)

# With rate parameter
expValLimWeibull(d = 2, shape = 2, rate = 0.2)

# With scale parameter
expValTruncWeibull(d = 2, shape = 2, scale = 5)

# With rate parameter
expValTruncWeibull(d = 2, shape = 2, rate = 0.2)

# Mean of values greater than d
expValTruncWeibull(d = 2, shape = 2, rate = 0.2, less.than.d = FALSE)

# With scale parameter
stopLossWeibull(d = 2, shape = 3, scale = 4)

# With rate parameter
stopLossWeibull(d = 2, shape = 3, rate = 0.25)

# With scale parameter
meanExcessWeibull(d = 2, shape = 3, scale = 4)

# With rate parameter
meanExcessWeibull(d = 2, shape = 3, rate = 0.25)

# With scale parameter
VatRWeibull(kap = .2, shape = 3, scale = 4)

# With rate parameter
VatRWeibull(kap = .2, shape = 3, rate = 0.25)

# With scale parameter
TVatRWeibull(kap = .2, shape = 3, scale = 4)

# With rate parameter
TVatRWeibull(kap = .2, shape = 3, rate = 0.25)

Binomial Distribution

Description

Binomial distribution with size n and probability of success p.

Usage

expValBinom(size, prob)

varBinom(size, prob)

expValTruncBinom(d, size, prob, less.than.d = TRUE)

VatRBinom(kap, size, prob)

TVatRBinom(kap, size, prob)

pgfBinom(t, size, prob)

mgfBinom(t, size, prob)

Arguments

Details

The binomial distribution with probability of success p for n trials has probability mass function :

Pr(X = k) = \left(\frac{n}{k}\right) p^n (1 - p)^{n - k}

for k = 0, 1, 2, \dots, n, p \in [0, 1], and n > 0

Value

Function :

• mgfBinom gives the moment generating function (MGF).

• pgfBinom gives the probability generating function (PGF).

• expValBinom gives the expected value.

• varBinom gives the variance.

• expValTruncBinom gives the truncated mean.

• TVatRBinom gives the Tail Value-at-Risk.

• VatRBinom gives the Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRBinom is a wrapper of the qbinom function from the stats package.

Examples

expValBinom(size = 3, prob = 0.5)

varBinom(size = 3, prob = 0.5)


expValTruncBinom(d = 2, size = 3, prob = 0.5)
expValTruncBinom(d = 0, size = 3, prob = 0.5, less.than.d = FALSE)


VatRBinom(kap = 0.8, size = 5, prob = 0.2)

TVatRBinom(kap = 0.8, size = 5, prob = 0.2)


pgfBinom(t = 1, size = 3, prob = 0.5)

mgfBinom(t = 1, size = 3, prob = 0.5)

Bivariate Ali-Mikhail-Haq Copula

Description

Computes CDF, PDF and simulations of of the bivariate Ali-Mikhail-Haq copula.

Usage

cBivariateAMH(u1, u2, dependencyParameter, ...)

cdBivariateAMH(u1, u2, dependencyParameter, ...)

crBivariateAMH(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The bivariate Ali-Mikhail-Haq copula has CDF :

Value

Function :

• cBivariateAMH returns the value of the copula.

• cdBivariateAMH returns the value of the density copula.

• crBivariateAMH returns simulated values of the copula.

Examples

cBivariateAMH(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

cdBivariateAMH(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

crBivariateAMH(numberSimulations = 10, seed = 42, dependencyParameter = 0.2)

Bivariate Cuadras-Augé Copula

Description

Computes CDF and simulations of the bivariate Cuadras-Augé copula.

Usage

cBivariateCA(u1, u2, dependencyParameter, ...)

crBivariateCA(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The bivariate Cuadras-Augé copula has CDF :

C(u_{1}, u_{2}) = u_{1}u_{2}^{1 - \alpha} \times% \textbf{1}_{\{u_{1} \leq u_{2}\}} + u_{1}^{1 - \alpha}u_{2} \times% \textbf{1}_{\{u_{1} \geq u_{2}\}}

for u_{1}, u_{2}, \alpha \in [0, 1]. It is the geometric mean of the independance and upper Fréchet bound copulas.

Value

Function :

• cBivariateCA returns the value of the copula.

• crBivariateCA returns simulated values of the copula.

Examples

cBivariateCA(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

crBivariateCA(numberSimulations = 10, seed = 42, dependencyParameter = 0.2)

Bivariate Clayton Copula

Description

Computes CDF, PDF and simulations of the bivariate Clayton copula.

Usage

cBivariateClayton(u1, u2, dependencyParameter, ...)

cdBivariateClayton(u1, u2, dependencyParameter, ...)

crBivariateClayton(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The bivariate Clayton copula has CDF :

Value

Function :

• cBivariateAMH returns the value of the copula.

• cdBivariateAMH returns the value of the density function associated to the copula.

Examples

cBivariateClayton(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

cdBivariateClayton(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

crBivariateClayton(numberSimulations = 10, seed = 42, dependencyParameter = 0.2)

Bivariate Eyraud-Farlie-Gumbel-Morgenstern (EFGM) Copula

Description

Computes CDF, PDF and simulations of the EFGM copula.

Usage

cBivariateEFGM(u1, u2, dependencyParameter)

cdBivariateEFGM(u1, u2, dependencyParameter)

crBivariateEFGM(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The EFGM copula has CDF :

Value

Function :

• cBivariateEFGM returns the value of the copula.

• cdBivariateEFGM returns the value of the density function associated to the copula.

• crBivariateEFGM returns simulated values of the copula.

Examples

cBivariateEFGM(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

cdBivariateEFGM(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

crBivariateEFGM(numberSimulations = 10, seed = 42, dependencyParameter = 0.2)

Bivariate Frank Copula

Description

Computes CDF, PDF and simulations of the bivariate Frank copula.

Usage

cBivariateFrank(u1, u2, dependencyParameter, ...)

cdBivariateFrank(u1, u2, dependencyParameter, ...)

crBivariateFrank(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The bivariate Frank copula has CDF :

Value

Function :

• cBivariateFrank returns the value of the copula.

• cdBivariateFrank returns the value of the density function associated to the copula.

• crBivariateFrank returns simulated values of the copula.

Examples

cBivariateFrank(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

cdBivariateFrank(u1 = .76, u2 = 0.4, dependencyParameter = 0.4)

crBivariateFrank(numberSimulations = 10, seed = 42, dependencyParameter = 0.2)

Bivariate Gumbel Copula

Description

Computes CDF, PDF and simulations of the bivariate Gumbel copula.

Usage

cBivariateGumbel(u1, u2, dependencyParameter, ...)

cdBivariateGumbel(u1, u2, dependencyParameter, ...)

crBivariateGumbel(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The bivariate Gumbel copula has CDF :

Value

Function :

• cBivariateGumbel returns the value of the copula.

• cdBivariateGumbel returns the value of the density function associated to the copula.

• crBivariateGumbel returns simulated values of the copula.

Examples

cBivariateGumbel(u1 = .76, u2 = 0.4, dependencyParameter = 1.4)

cdBivariateGumbel(u1 = .76, u2 = 0.4, dependencyParameter = 1.4)

crBivariateGumbel(numberSimulations = 10, seed = 42, dependencyParameter = 1.2)

Bivariate Marshall-Olkin Copula

Description

Computes CDF and simulations of the bivariate Marshall-Olkin copula.

Usage

cBivariateMO(u1, u2, dependencyParameter, ...)

crBivariateMO(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The bivariate Marshall-Olkin copula has CDF :

C(u_{1}, u_{2}) = u_{1}u_{2}^{1 - \beta} \times% \textbf{1}_{\{u_{1}^{\alpha} \leq u_{2}^{\beta}\}} + % u_{1}^{1 - \alpha}u_{2} \times \textbf{1}_{\{u_{1}^{\alpha}% \geq u_{2}^{\beta}\}}

for u_{1}, u_{2}, \alpha, \beta \in [0, 1]. It is the geometric mean of the independance and upper Fréchet bound copulas.

Value

Function :

• cBivariateMO returns the value of the copula.

• crBivariateMO returns simulated values of the copula.

Examples

cBivariateMO(u1 = .76, u2 = 0.4, dependencyParameter = c(0.4, 0.3))

crBivariateMO(numberSimulations = 10, seed = 42, dependencyParameter = c(0.2, 0.5))

Erlang Distribution

Description

Erlang distribution with shape parameter n and rate parameter \beta.

Usage

dErlang(x, shape, rate = 1/scale, scale = 1/rate)

pErlang(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE)

expValErlang(shape, rate = 1/scale, scale = 1/rate)

varErlang(shape, rate = 1/scale, scale = 1/rate)

kthMomentErlang(k, shape, rate = 1/scale, scale = 1/rate)

expValLimErlang(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncErlang(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE)

stopLossErlang(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessErlang(d, shape, rate = 1/scale, scale = 1/rate)

VatRErlang(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRErlang(kap, shape, rate = 1/scale, scale = 1/rate)

mgfErlang(t, shape, rate = 1/scale, scale = 1/rate)

Arguments

Details

The Erlang distribution with shape parameter n and rate parameter \beta has density:

f\left(x\right) = \frac{\beta^{n}}{\Gamma(n)} x^{n - 1}% \mathrm{e}^{-\beta x}

for x \in \mathcal{R}^+, \beta > 0, n \in \mathcal{N}^+.

Value

Function :

• dErlang gives the probability density function (PDF).

• pErlang gives the cumulative density function (CDF).

• expValErlang gives the expected value.

• varErlang gives the variance.

• kthMomentErlang gives the kth moment.

• expValLimErlang gives the limited mean.

• expValTruncErlang gives the truncated mean.

• stopLossErlang gives the stop-loss.

• meanExcessErlang gives the mean excess loss.

• VatRErlang gives the Value-at-Risk.

• TVatRErlang gives the Tail Value-at-Risk.

• mgfErlang gives the moment generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Note

Function VatRErlang is a wrapper of the qgamma function from the stats package.

Examples

dErlang(x = 2, shape = 2, scale = 4)

pErlang(q = 2, shape = 2, scale = 4)

expValErlang(shape = 2, scale = 4)

varErlang(shape = 2, scale = 4)

kthMomentErlang(k = 3, shape = 2, scale = 4)

expValLimErlang(d = 2, shape = 2, scale = 4)

# With rate parameter
expValTruncErlang(d = 2, shape = 2, scale = 4)

# Values greater than d
expValTruncErlang(d = 2, shape = 2, scale = 4, less.than.d = FALSE)

stopLossErlang(d = 2, shape = 2, scale = 4)

meanExcessErlang(d = 3, shape = 2, scale = 4)

# With scale parameter
VatRErlang(kap = .2, shape = 2, scale = 4)

# With rate parameter
VatRErlang(kap = .2, shape = 2, rate = 0.25)

# With scale parameter
TVatRErlang(kap = .2, shape = 3, scale = 4)

# With rate parameter
TVatRErlang(kap = .2, shape = 3, rate = 0.25)

mgfErlang(t = 2, shape = 2, scale = .25)

Fréchet Copula

Description

Computes CDF and simulations of the Fréchet copula.

Usage

cFrechet(u1, u2, dependencyParameter, ...)

crFrechet(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

Details

The Fréchet copula has CDF :

C(u_{1}, u_{2}) = (1 - \alpha - \beta)(u_{1} \times u_{2}) + \alpha\min(u_{1}, u_{2})% + \beta\max(u_{1} + u_{2} - 1, 0)

for u_{1}, u_{2}, \alpha, \beta \in [0, 1] and \alpha + \beta \leq 1.

Value

Function :

• cFrechet returns the value of the copula.

• crFrechet returns simulated values of the copula.

Examples

cFrechet(u1 = .76, u2 = 0.4, dependencyParameter = c(0.2, 0.3))

crFrechet(numberSimulations = 10, seed = 42, dependencyParameter = c(0.2, 0.3))

Fréchet Lower Bound Copula

Description

Computes CDF and simulations of the Fréchet lower bound copula.

Usage

cFrechetLowerBound(u1, u2, ...)

crFrechetLowerBound(numberSimulations = 10000, seed = 42)

Arguments

Details

The Fréchet lower bound copula has CDF :

C(u_{1}, u_{2}) = \max(u_{1} + u_{2} - 1, 0)

for u_{1}, u_{2} \in [0, 1].

Value

Function :

• cFrechetLowerBound returns the value of the copula.

• crFrechetLowerBound returns simulated values of the copula.

Examples

cFrechetLowerBound(u1 = .76, u2 = 0.4)

crFrechetLowerBound(numberSimulations = 10, seed = 42)

Fréchet Upper Bound Copula

Description

Computes CDF and simulations of the Fréchet upper bound copula.

Usage

cFrechetUpperBound(u1, u2, ...)

crFrechetUpperBound(numberSimulations = 10000, seed = 42)

Arguments

Details

The Fréchet upper bound copula has CDF :

C(u_{1}, u_{2}) = \min(u_{1}, u_{2})

for u_{1}, u_{2} \in [0, 1].

Value

Function :

• cFrechetUpperBound returns the value of the copula.

• crFrechetUpperBound returns simulated values of the copula.

Examples

cFrechetUpperBound(u1 = .56, u2 = 0.4)

crFrechetUpperBound(numberSimulations = 10, seed = 42)

Independence Copula

Description

Computes CDF and simulations of the independence copula.

Usage

cIndependent(u1, u2, ...)

crIndependent(numberSimulations = 10000, seed = 42)

Arguments

Details

The independence copula has CDF :

C(u_{1}, u_{2}) = u_{1} \times u_{2}

for u_{1}, u_{2} \in [0, 1].

Value

Function :

• cIndependent returns the value of the copula.

• crIndependent returns simulated values of the copula.

Examples

cIndependent(u1 = .76, u2 = 0.4)

crIndependent(numberSimulations = 10, seed = 42)

Loglogistic Distribution

Description

Loglogistic distribution with shape parameter \tau and scale parameter \lambda.

Usage

dLlogis(x, shape, rate = 1/scale, scale = 1/rate)

pLlogis(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE)

expValLlogis(shape, rate = 1/scale, scale = 1/rate)

varLlogis(shape, rate = 1/scale, scale = 1/rate)

kthMomentLlogis(k, shape, rate = 1/scale, scale = 1/rate)

expValLimLlogis(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncLlogis(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE)

stopLossLlogis(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessLlogis(d, shape, rate = 1/scale, scale = 1/rate)

VatRLlogis(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRLlogis(kap, shape, rate = 1/scale, scale = 1/rate)

Arguments

Details

The loglogistic distribution with shape parameter \tau and scale parameter \lambda has density:

\frac{\tau \lambda^\tau x^{\tau -1}}{(\lambda^{\tau }+x^{\tau })^{2}}

for x \in \mathcal{R}^+, \lambda, \tau > 0.

Value

Function :

• dLlogis gives the probability density function (PDF).

• pLlogis gives the cumulative density function (CDF).

• expValLlogis gives the expected value.

• varLlogis gives the variance.

• kthMomentLlogis gives the kth moment.

• expValLimLlogis gives the limited mean.

• expValTruncLlogis gives the truncated mean.

• stopLossLlogis gives the stop-loss.

• meanExcessLlogis gives the mean excess loss.

• VatRLlogis gives the Value-at-Risk.

• TVatRLlogis gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

dLlogis(x = 2, shape = 2, scale = 4)

# With scale parameter
pLlogis(q = 3, shape = 3, scale = 5)

# With rate parameter
pLlogis(q = 3, shape = 3, rate = 0.2)

# Survival function
pLlogis(q = 3, shape = 3, rate = 0.2, lower.tail = FALSE)

expValLlogis(shape = 2, scale = 4)

varLlogis(shape = 3, scale = 4)

kthMomentLlogis(k = 3, shape = 5, scale = 4)

expValLimLlogis(d = 2, shape = 2, scale = 4)

# With rate parameter
expValTruncLlogis(d = 2, shape = 2, scale = 4)

# Values greater than d
expValTruncLlogis(d = 2, shape = 2, scale = 4, less.than.d = FALSE)

stopLossLlogis(d = 2, shape = 2, scale = 4)

meanExcessLlogis(d = 3, shape = 2, scale = 4)

# With scale parameter
VatRLlogis(kap = .2, shape = 2, scale = 4)

# With rate parameter
VatRLlogis(kap = .2, shape = 2, rate = 0.25)

# With scale parameter
TVatRLlogis(kap = .2, shape = 3, scale = 4)

# With rate parameter
TVatRLlogis(kap = .2, shape = 3, rate = 0.25)

Discrete Uniform Distribution

Description

Discrete uniform distribution with min a and max b.

Usage

pUnifD(q, min = 0, max = 1)

dUnifD(x, min = 0, max = 1)

varUnifD(min = 0, max = 1)

expValUnifD(min = 0, max = 1)

Arguments

Details

The (discrete) uniform distribution with min and max parameters a and b respectively has density:

\textrm{Pr}\left(X = x \right) = \frac{1}{b - a + 1}

for x \in \{a, a + 1, \dots, b - 1, b\}.

Value

Function :

• dUnifD gives the probability density function (PDF).

• pUnifD gives the cumulative density function (CDF).

• expValUnifD gives the expected value.

• varUnifD gives the variance.

Invalid parameter values will return an error detailing which parameter is problematic.

Examples

pUnifD(q = 0.2, min = 0, max = 1)

dUnifD(min = 0, max = 1)

varUnifD(min = 0, max = 1)

expValUnifD(min = 0, max = 1)