Univariate Pseudo-Random Number Generation

Description

This package implements the algorithms described in Demirtas (2005) for pseudo-random number generation of 17 univariate distributions. The following distributions are available: Left Truncated Gamma, Laplace, Inverse Gaussian, Von Mises, Zeta (Zipf), Logarithmic, Beta-Binomial, Rayleigh, Pareto, Non-central t, Non-central Chi-squared, Doubly non-central F, Standard t, Weibull, Gamma with \alpha<1, Gamma with \alpha>1, and Beta with \alpha<1 and \beta<1. For some distributions, functions that have similar capabilities exist in the base package; the functions herein should be regarded as complementary tools.

The methodology for each random-number generation procedure varies and each distribution has its own function. draw.left.truncated.gamma, draw.von.mises, draw.inverse.gaussian, draw.zeta, draw.gamma.alpha.less.than.one, and draw.beta.alphabeta.less.than.one are based on acceptance/rejection region techniques. draw.rayleigh, draw.pareto, and draw.weibull utilize the inverse CDF method. The chop-down method is used for draw.logarithmic. In draw.laplace, a sample from an exponential distribution with mean 1/\lambda is generated and subsequently the sign is changed with probability 0.5 and all variables are shifted by \alpha. For the Beta-Binomial distribution in draw.beta.binomial, \pi is generated as the appropriate \beta and used as the success probability for the binomial portion. draw.noncentral.t utilizes on arithmetic functions of normal and chi-squared random variables. draw.noncentral.chisquared is based on the sum of squared random normal variables, and draw.noncentral.F is a ratio of chi-squared random variables generated via draw.noncentral.chisquared. draw.t employs a rejection polar method developed by Bailey (1994). draw.gamma.alpha.greater.than.one uses a ratio of uniforms method by Cheng and Feast (1979).

Details

Author(s)

Hakan Demirtas, Rawan Allozi, Ran Gao

Maintainer: Ran Gao <rgao8@uic.edu>

References

Bailey, R. W. (1994). Polar generation of random variates with the t-distribution. Mathematics of Computation, 62, 779-781.

Cheng, R. C. H., & Feast, G. M. (1979). Some simple gamma variate generation. Applied Statistics, 28, 290-295.

Demirtas, H. (2005). Pseudo-random number generation in R for some univariate distributions. Journal of Modern Applied Statistical Methods, 4(1), 300-311.

Generates variates from Beta distribution with max(\alpha,\beta)<1

Description

This function implements pseudo-random number generation for a Beta distribution for \max(\alpha,\beta)<1 with pdf

f(x|\alpha,\beta)=\frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}

for 0 \leq x \leq 1, 0 < \alpha < 1, and 0 < \beta < 1 where \alpha and \beta are the shape parameters and B(\alpha,\beta) is the complete beta function.

Usage

draw.beta.alphabeta.less.than.one(nrep,alpha,beta)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Jhonk, M. D. (1964). Erzeugung von betaverteilter und gammaverteilter zufallszahlen. Metrika, 8, 5-15.

Examples

draw.beta.alphabeta.less.than.one(nrep=100000,alpha=0.7,beta=0.4)

Generates variates from Beta-binomial distribution

Description

This function implements pseudo-random number generation for a Beta-binomial distribution with pmf

f(x|n,\alpha,\beta) = \frac{n!}{x!(n-x)!B(\alpha,\beta)}\int_{0}^{1}\pi^{\alpha-1+x}(1-\pi)^{n+\beta-1-x}d\pi

for x=0,1,2,..., \alpha>0, and \beta>0, where n is the sample size, \alpha and \beta are the shape parameters and B(\alpha,\beta) is the complete beta function.

Usage

draw.beta.binomial(nrep,alpha,beta,n)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.beta.binomial(nrep=100000,alpha=0.2,beta=0.25,n=10)

draw.beta.binomial(nrep=100000,alpha=2,beta=3,n=10)

draw.beta.binomial(nrep=100000,alpha=600,beta=400,n=20)

Generates variation from Gamma distribution with \alpha>1

Description

This function implements pseudo-random number generation for a Gamma distribution for \alpha>1 with pdf

f(x|\alpha,\beta)=\frac{1}{\Gamma(\alpha)\beta^{\alpha}}x^{\alpha-1}e^{-x/\beta}

for 0 \leq x < \infty and \min(\alpha,\beta)>0 where \alpha and \beta are the shape and scale parameters, respectively.

Usage

draw.gamma.alpha.greater.than.one(nrep,alpha,beta)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Cheng, R. C. H., & Feast, G. M. (1979). Some simple gamma variate generation. Applied Statistics, 28, 290-295.

Examples

draw.gamma.alpha.greater.than.one(nrep=100000,alpha=2,beta=2)

draw.gamma.alpha.greater.than.one(nrep=100000,alpha=3,beta=0.4)

Generates variation from Gamma distribution with \alpha<1

Description

This function implements pseudo-random number generation for a gamma distribution for \alpha<1 with pdf

f(x|\alpha,\beta)=\frac{1}{\Gamma(\alpha)\beta^{\alpha}}x^{\alpha-1}e^{-x/\beta}

for 0 \leq x < \infty and \min(\alpha,\beta)>0 where \alpha and \beta are the shape and scale parameters, respectively.

Usage

draw.gamma.alpha.less.than.one(nrep,alpha,beta)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Ahrens, J. H., & Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial distributions. Computing, 1, 223-246.

Examples

draw.gamma.alpha.less.than.one(nrep=100000,alpha=0.5,beta=2)

Generates variation from inverse Gaussian distribution

Description

This function implements pseudo-random number generation for an inverse Gaussian distribution with pdf

f(x|\mu,\lambda)=(\frac{\lambda}{2\pi})^{1/2}x^{-3/2}e^{-\frac{\lambda(x-\mu)^2}{2\mu^2x}}

for x > 0, \mu > 0, and \lambda > 0 where \mu and \lambda are the location and scale parameters, respectively.

Usage

draw.inverse.gaussian(nrep,mu,lambda)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Michael, J. R., William, R. S., & Haas, R. W. (1976). Generating random variates using transformations with multiple roots. The American Statistician, 30, 88-90.

Examples

draw.inverse.gaussian(nrep=100000,mu=1,lambda=1)

draw.inverse.gaussian(nrep=100000,mu=3,lambda=1)

Generates variates from Laplace distribution

Description

This function implements pseudo-random number generation for a Laplace (double exponential) distribution with pdf

f(x|\lambda,\alpha)=\frac{\lambda}{2}e^{-\lambda|x-\alpha|}

for \lambda>0 where \alpha and \lambda are the location and scale parameters, respectively.

Usage

draw.laplace(nrep, alpha, lambda)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.laplace(nrep=100000, alpha=4, lambda=2)

draw.laplace(nrep=100000, alpha=-5, lambda=4)

Generates variates from left truncated Gamma distribution

Description

This function implements pseudo-random number generation for a left-truncated gamma distribution with pdf

f(x|\alpha, \beta) = \frac{1}{(\Gamma(\alpha)-\Gamma_{\tau/\beta}(\alpha))\beta^{\alpha}}x^{\alpha-1}e^{-x/\beta}

for 0<\tau\leq x, and min(\tau, \beta)>0 where \alpha and \beta are the shape and scale parameters, respectively, \tau is the cutoff point at which truncation occurs, and \Gamma_{\tau/\beta} is the incomplete gamma function.

Usage

draw.left.truncated.gamma(nrep,alpha,beta,tau)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Dagpunar, J. S. (1978). Sampling of variates from a truncated gamma distribution. Journal of Statistical Computation and Simulation, 8, 59-64.

Examples

draw.left.truncated.gamma(nrep=100000,alpha=5,beta=1,tau=0.5)

draw.left.truncated.gamma(nrep=100000,alpha=2,beta=2,tau=0.1)

Generates variates from logarithmic distribution

Description

This function implements pseudo-random number generation for a logarithmic distribution with pmf

f(x|\theta) = - \frac{\theta^{x}}{x\log(1-\theta)}

for x=1,2,3,... and 0 < \theta < 1.

Usage

draw.logarithmic(nrep,theta)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Kemp, A. W. Efficient generation of logarithmically distributed pseudo-random variables. Applied Statistics, 30, 249-253.

Examples

draw.logarithmic(nrep=100000,theta=0.33)

draw.logarithmic(nrep=100000,theta=0.66)

Generates variates from doubly non-central F distribution

Description

This function implements pseudo-random number generation for a doubly non-central F distribution

F=\frac{X_{1}^{2}/n}{X_{2}^{2}/m}

where X_{1}^{2}\sim \chi^{2}(n,\lambda_{1}), X_{2}^{2}\sim \chi^{2}(m,\lambda_{2}), n and m are numerator and denominator degrees of freedom, respectively, and \lambda_{1} and \lambda_{2} are the numerator and denominator non-centrality parameters, respectively. It includes central and singly non-central F distributions as a special case.

Usage

draw.noncentral.F(nrep,dof1,dof2,ncp1,ncp2)

Arguments

Value

A vector containing generated data.

See Also

draw.noncentral.chisquared

Examples

draw.noncentral.F(nrep=100000,dof1=2,dof2=4,ncp1=2,ncp2=4)

Generates variates from non-central chi-squared distribution

Description

This function implements pseudo-random number generation for a non-central chi-squared distribution with pdf

f(x|\lambda,\nu)=\frac{e^{-(x+\lambda)/2}x^{\nu/2-1}}{2^{\nu/2}} \sum_{k=0}^{\infty} \frac{(\lambda x)^{k}}{4^{k}k!\Gamma(k+\nu/2)}

for 0 \leq x < \infty, \lambda>0, and \nu>1, where \lambda is the non-centrality parameter and \nu is the degrees of freedom.

Usage

draw.noncentral.chisquared(nrep,dof,ncp)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.noncentral.chisquared(nrep=100000,dof=2,ncp=1)

draw.noncentral.chisquared(nrep=100000,dof=5,ncp=2)

Generates variates from doubly non-central t distribution

Description

This function implements pseudo-random number generation for a non-central t distribution

\frac{Y}{\sqrt{U/\nu}}

where U is a central chi-square random variable with \nu degrees of freedom and Y is an independent, normally distributed random variable with variance 1 and mean \lambda.

Usage

draw.noncentral.t(nrep,nu,lambda)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.noncentral.t(nrep=100000,nu=4,lambda=2)

draw.noncentral.t(nrep=100000,nu=5,lambda=1)

Generates variates from Pareto distribution

Description

This function implements pseudo-random number generation for a Pareto distribution with pdf

f(x|\alpha,\beta)=\frac{ab^{a}}{x^{a+1}}

for 0 < b \leq x < \infty and a>0 where a and b are the shape and location parameters, respectively.

Usage

draw.pareto(nrep,shape,location)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.pareto(nrep=100000,shape=11,location=11)

draw.pareto(nrep=100000,shape=8,location=10)

Generates variates from Rayleigh distribution

Description

This function implements pseudo-random number generation for a Rayleigh distribution with pdf

f(x|\sigma)=\frac{x}{\sigma^2}e^{-x^2/2\sigma^2}

for x \geq 0 and \sigma > 0 where \sigma is the scale parameter.

Usage

draw.rayleigh(nrep,sigma)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.rayleigh(nrep=100000,sigma=0.5)

draw.rayleigh(nrep=100000,sigma=3)

Generates variates from standard t distribution

Description

This function implements pseudo-random number generation for a standard-t distribution with pdf

f(x|\nu)=\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\nu\pi}}(1+\frac{x^2}{\nu})^{-(\nu+1)/2}

for -\infty < x < \infty where \nu is the degrees of freedom.

Usage

draw.t(nrep,dof)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Bailey, R. W. (1994). Polar generation of random variates with the t-distribution. Mathematics of Computation, 62, 779-781.

Examples

draw.t(nrep=100000,dof=2)

draw.t(nrep=100000,dof=6)

Generates variates from Von Mises distribution

Description

This function implements pseudo-random number generation for a Von Mises distribution with pdf

f(x|K)=\frac{1}{2\pi I_{0}(K)}e^{Kcos(x)}

for -\pi \leq x \leq \pi and K > 0 where I_{0}(K) is a modified Bessel function of the first kind of order 0.

Usage

draw.von.mises(nrep,K)

Arguments

Value

A list of length three containing generated data, the theoretical mean, and the empirical mean with names y, theo.mean, and emp.mean, respectively.

References

Best, D. J., & Fisher, N. I. (1979). Efficient simulation of the von mises distribution. Applied Statistics, 28, 152-157.

Examples

draw.von.mises(nrep=100000,K=10)

draw.von.mises(nrep=100000,K=0.5)

Generates variates from Weibull distribution

Description

This function implements pseudo-random number generation for a Weibull distribution with pdf

f(x|\alpha,\beta)=\frac{\alpha}{\beta^{\alpha}}x^{\alpha-1}e^{-(x/\beta)^{\alpha}}

for 0 \leq x < \infty and \min(\alpha,\beta)>0 where \alpha and \beta are the shape and scale parameters, respectively.

Usage

draw.weibull(nrep, alpha, beta)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

Examples

draw.weibull(nrep=100000, alpha=0.5, beta=1)

draw.weibull(nrep=100000, alpha=5, beta=1)

Generates variates from Zeta (Zipf) distribution

Description

This function implements pseudo-random number generation for a Zeta (Zipf) distribution with pmf

f(x|\alpha)=\frac{1}{\zeta(\alpha)x^{\alpha}}

for x=1,2,3,... and \alpha > 1 where \zeta(\alpha)=\sum_{x=1}^{\infty} x^{-\alpha}.

Usage

draw.zeta(nrep, alpha)

Arguments

Value

A list of length five containing generated data, the theoretical mean, the empirical mean, the theoretical variance, and the empirical variance with names y, theo.mean, emp.mean, theo.var, and emp.var, respectively.

References

Devroye, L. (1986). Non-Uniform random variate generation. New York: Springer-Verlag.

Examples

draw.zeta(nrep=100000,alpha=4)