CRAN Task View: Probability Distributions

For most of the classical distributions, base R provides probability distribution functions (p), density functions (d), quantile functions (q), and random number generation (r). Beyond this basic functionality, many CRAN packages provide additional useful distributions. In particular, multivariate distributions as well as copulas are available in contributed packages. If you think information is not accurate or not complete, please let me know.

Base functionality:

• Base R provides probability distribution functions density functions , quantile functions , and random number generation where foo indicates the type of distribution: beta ( foo = beta), binomial binom, Cauchy cauchy, chi-squared chisq, exponential exp, Fisher F f, gamma gamma, geometric geom, hypergeometric hyper, logistic logis, lognormal lnorm, negative binomial nbinom, normal norm, Poisson pois, Student t t, uniform unif, Weibull weibull. Following the same naming scheme, but somewhat less standard are the following distributions in base R: probabilities of coincidences (also known as "birthday paradox") birthday (only p and q), studentized range distribution tukey (only p and q), Wilcoxon signed rank distribution signrank, Wilcoxon rank sum distribution wilcox.

Discrete distributions:

• Basic distributions : The binomial distribution (in particular Bernoulli distribution) is already implemented in base R. The discrete uniform distribution can be easily obtained with the basic functions. Dirac distribution is provided by distr. Truncated versions of the binomial and Poisson distributions as well as zero-inflated versions of the binomial, Poisson and negative binomial distributions are implemented in VGAM.
• Conway-Maxwell-Poisson distribution : This can be found in compoisson.
• Hypergeometric distributions : Extented hypergeometric distribution can be found in BiasedUrn package, which provides not only p, d, q, r functions but also mean, variance, mode functions.
• Zipf law : Package zipfR provides tools for the Zipf and the Zipf-Mandelbrot distributions. VGAM also implements the Zipf distribution.
• Further distributions : The VGAM package provides several additional distributions, namely: Logarithmic, Skellam, Yule-Simon, Zeta and Haight's Zeta, Borel-Tanner and Felix distribution.

Continuous distributions:

• Arcsine distribution : implemented in package distr.
• Beta distribution and its extensions : Base R provides the d, p, q, r functions for this distribution (see above). actuar provides moments and limited expected values. It also provide the d, p, q, r functions for the generalized beta and the inverse tranformed beta distribution. The beta of the second kind and the generalized beta distribution can be found in VGAM. Several special cases of the generalized beta distribution are also implemented in VGAM: Lomax, inverse Lomax, Dagum, Fisk (aka log logistic), (inverse or not) paralogistic and Singh-Maddala distribution.
• Benini distribution : Provided in VGAM.
• Birnbaum-Saunders distribution : Provided in packages bs and VGAM. The generalized Birnbaum-Saunders distribution is implemented in gbs.
• Cardioid distribution : Provided in VGAM.
• Cauchy distribution : Base R provides the d, p, q, r functions for this distribution (see above). Another implementation is available in lmomco.
• Chi(-squared or not) distributions : Base R provides the d, p, q, r functions for this distribution (see above). Moments, limited expected values and the moment generating function are provided by actuar. The inverse chi-squared distribution does not seem to be available. The Chi distribution and the non-central Chi distribution are not yet implemented.
• Davies distribution : The Davies distribution is provided in Davies package.
• Dirichlet distribution : functions d, r are provided in MCMCpack and bayesm.
• Exponential distribution and its extensions : Base R provides the d, p, q, r functions for this distribution (see above). actuar provides additional functions such as the moment generating function, moments and limited expected values. It also has the d, p, q, r for the inverse exponential distribution. The shifted exponential distribution is implemented in lmomco package with d, p, q, r functions.
• Frechet distribution : Provided in VGAM and evd.
• Gamma distribution and its extensions : Base R provides the d, p, q, r functions for this distribution (see above). actuar provides d, p, q, r functions for the inverse, the inverse transformed and the log gamma distributions while ghyp provides those functions for the variance gamma distribution. VGAM provides d, p, q, r functions of the log gamma and the generalized gamma distribution.
• Gaussian (or normal) distribution and its extensions : Base R provides the d, p, q, r functions for this distribution (see above). actuar provides the moment generating function and moments. ghyp package provides d, p, q, r functions for the raw and the generalized inverse gaussian distribution, in the context of the generalized hyperbolic distribution. SuppDists provides functions for the inverse Gaussian distribution as well and furthermore includes functions for computing moments, skewness, kurtosis. VGAM also implements the folded and the skewed normal distribution, the inverse gaussian distribution. lmomco implements the generalized normal distribution. The log normal distribution is implemented in Base R (see above). Finally, the multivariate Gaussian distribution is provided by the packages mvtnorm and mnormt, while mvtnormpcs implements multivariate student/normal integrals, given a correlation matrix structure.
• General Error Distribution (also known as exponential power distribution) : provided in normalp.
• Generalized Extreme Value distribution : Provided in lmomco (d, p, q) , VGAM and evd (d, p, q, r). Both bivariate and multivariate Extreme Value distributions as well as order/maxima/minima distributions are implemented in evd (d, p, r).
• Gumbel distribution : Provided in packages lmomco, VGAM and evd. The reverse Gumbel distribution is implemented in lmomco.
• Kappa distribution : Provided in lmomco.
• Kumaraswamy distribution : Provided in package VGAM.
• Lambda distribution : The Tukey lambda (also known as the generalized lambda) distribution is implemented in packages gld and Davies.
• Laplace and asymetric Laplace distribution : Provided in VGAM. Laplace distribution (also called double exponential distribution) is implemented in distr.
• Logistic distributions and its extensions : Base R provides the d, p, q, r functions for this distribution (see above). actuar provides d, p, q, r functions for the log logistic (also called Fisk), the paralogistic and the inverse paralogistic distributions. VGAM package also implements these distributions plus the bivariate logistic distribution. lmomco implements the generalized logistic distribution.
• Maxwell distribution : Provided in VGAM.
• Nakagami distribution : Provided in VGAM.
• Pareto distribution : d, p, q, r functions are implemented in VGAM for the Pareto distribution type IV (which includes Burr's distribution, Pareto type III, Pareto type II (also called the Generalized Pareto Distribution - GPD) and Pareto type I) and the (upper/lower) truncated Pareto distribution. In an actuarial context, actuar provides d, p, q, r functions as well as moments and limited expected values for the Pareto, the inverse Pareto, the 'generalized pareto' distributions, the Burr and the inverse Burr distributions. Finally lmomco, POT and evd packages implement the GPD.
• Pearson type III distribution : Available in lmomco.
• Phase-type distributions : Provided in actuar.
• Rayleigh distribution : Provided in packages VGAM and lmomco.
• Rice distribution : Provided in VGAM.
• Slash distribution : Provided in VGAM.
• Student distribution and its extensions : Base R provides the d, p, q, r functions for Student and non central Student distribution (see above). The skewed Student distribution is provided by skewt and sn packages. Package tdist provides d, p, q functions for the distribution of a linear combination of independent Student distributions. The multivariate Student distribution is provided by the packages mvtnorm and mnormt.
• Triangle distribution : Provided in packages triangle and VGAM.
• Tweedie distribution : the Tweedie distribution is implemented in package tweedie. Let us note that the Tweedie distribution is not necessarily continuous, a special case of it is the Poisson distribution.
• Uniform distribution : d, p, q, r functions are of course provided in R. See section RNG for random number generation topics.
• Wakeby distribution : Provided in lmomco.
• Weibull distribution and its extensions : Base R provides the d, p, q, r functions for this distribution (see above). The inverse Weibull is provided by actuar package and also the moments and the limited expevted value for both the raw and the inverse Weibull distribution. Finally, lmomco implements the Weibull distribution while evd implements the reverse Weibull distribution.
• Wishart and inverse Wishart distributions : functions d, r are provided in MCMCpack and bayesm.

Mixture of probability laws:

• Gaussian mixture : Functions d, r are provided by mixtools package when dealing with finite mixture models.
• Gamma mixture : Gamma shape mixtures are implemented (d, p, r) in the GSM package.
• Generic mixtures : there is an implementation via S4-class UnivarMixingDistribution in package distr.
• Student mixture : The AdMit package provides d, r functions for Student mixtures in the context of Adaptive Mixture of Student-t distributions.

Copulas:

• Unified approaches : The packages fCopulae, copula provide a lot of general functionality for copulas.
• Archimedean copulas : The Frank bivariate distribution is available in VGAM. fCopulae implements the 22 Archimedean copulas of Nelsen (1998, Introduction to Copulas , Springer-Verlag) including Gumbel, Frank, Clayton, and Ali-Mikhail-Haq. gumbel is a standalone package for the Gumbel copula and VGAM provides the Ali-Mikhail-Haq bivariate distribution. Generalized Archimedean copulas are implemented in the fgac package.
• Cubic copula : Not yet implemented
• Dirichlet copula : Not yet implemented
• Elliptical copulas : Gaussian, Student and Cauchy copulas are implemented in fCopulae for the bivariate cases. copula provides the Gaussian and the Student copula.
• Extreme value copulas : fCopulae provides the following copulas Gumbel, Galambos, Husler-Reiss, Tawn, or BB5. copula also implements Gumbel, Galambos and Husler-Reiss.
• Eyraud-Farlie-Gumbel-Morgenstern : Provided in VGAM and copula.
• Mardia copula : Not yet implemented
• Plackett : Provided in VGAM and copula.

Random Number Generators:

• Basic functionality: R provides several random number generators (RNGs). The random seed can be provided via set.seed and the kind of RNG can be specified using RNGkind. The default RNG is the Mersenne-Twister algorithm. Other generators include Wichmann-Hill, Marsaglia-Multicarry, Super-Duper, Knuth-TAOCP, Knuth-TAOCP-2002, as well as user-supplied RNGs. For normal random numbers, the following algorithms are available: Kinderman-Ramage, Ahrens-Dieter, Box-Muller, Inversion (default).
• Setting the RNG: In addition to the tools above, setRNG provides an easy way to set RNG.
• Pseudo-randomness : RDieHarder offers several dozen new RNGs from the GNU GSL and randtoolbox provides more recent RNGs such as WELL and SF Mersenne-Twister. rsprng implements Scalable Parallel RNGs library, whereas randaes provides the deterministic part of the Fortuna cryptographic pseudorandom number generator (AES). Furthermore, rstream focuses on multiple independent streams of random numbers from different sources (in a object oriented approach). For non-uniform generation, the Runuran package interfaces to the UNU.RAN library for universal non-uniform RNGs, while the rv package provides a class to easily manipulate and summarize simulations in R.
• Quasi-randomness : The randtoolbox provides the following quasi random sequences: the Sobol sequence, the Halton (hence Van Der Corput) sequence and the Torus sequence (also known as Kronecker sequence). lhs package implements the latin hypercube sampling, an hybrid quasi/pseudo random method.
• True randomness : The random package provides several functions that access the true random number service at random.org .
• RNG tests : RDieHarder offers numerous tests of RNGs based on a reimplementation and extension of Marsaglia's DieHarder battery. randtoolbox provides basic RNG tests.

Miscellaneous:

• Hierarchical models : Distributions whose some parameters are no longer constant but random according to a particular distribution. VGAM provides a lot of hierarchical models: beta/binomial, beta/geometric and beta/normal distributions. bayesm implements: binary logit, linear, multivariate logit and negative binomial models. Furthermore LearnBayes and MCMCpack provides poisson/gamma, beta/binomial, normal/normal and multinomial/Dirichlet models.
• Transformation : Lebesgue decomposition are implemented in distr, as well as Convolution, Truncation and Huberization of distributions. Furthermore, distr provides distribution of the maximum or minimum of two distributions. See Object-orientation below.
• Object-orientation : General discrete and continuous distributions are implemented in package distr respectively via S4-class DiscreteDistribution and AbscontDistribution providing the classic d, p, q and r functions. distrEx extends available distributions to multivariate and conditional distributions as well as methods to compute useful statistics (expectation, variance,...) and distances between distributions (Hellinger, Kolmogorov,... distance). Finally package distrMod provides functions for the computation of minimum criterion estimators (maximum likelihood and minimum distance estimators). See other packages of the distr-family (distrSim, distrTEst, distrTeach, distrDoc).
• TI 83 scientific calculator : distributions provides the base R probability distributions (binomial, poisson, geometric, normal, chi square, Fisher, Student) based on TI-83 Plus graphic scientific calculator.
• Transversal functions : Package modeest provides mode estimation for various distributions, while lmomco focuses on moments estimation. VGAM provides a lot of parameter estimation for usual and "exotic" distributions. Package MASS (in the VR bundle) implements the flexible fitdistr function for parameter estimations. Bivariate copulas can be fitted with mlCopulaSelection.

CRAN packages:

• actuar (core)
• AdMit
• bayesm
• BiasedUrn
• bs
• compoisson
• copula (core)
• Davies
• distr (core)
• distrDoc
• distrEx
• distributions
• distrMod
• distrSim
• distrTeach
• distrTEst
• evd
• fCopulae (core)
• fgac
• gbs
• ghyp
• gld
• GLDEX
• GSM
• gumbel
• HyperbolicDist
• LearnBayes
• lhs
• lmomco (core)
• MCMCpack
• mixtools
• mlCopulaSelection
• mnormt (core)
• mvtnorm (core)
• mvtnormpcs
• normalp
• POT
• randaes
• random
• randtoolbox
• RDieHarder
• rsprng
• rstream
• Runuran
• rv
• setRNG
• skewt
• sn
• SuppDists (core)
• tdist
• triangle
• tweedie
• VGAM (core)
• zipfR

Related links:

• actuar: an R package for actuarial science
• fCopulae package, part of the Rmetrics project
• Journal of Statistical Software: R Programs for Truncated Distributions
• The "distrXXX"-Family of R-Packages