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.

The maintainers gratefully acknowledge Achim Zeileis, David Luethi, Tobias Verbeke, Robin Hankin, Mathias Kohl, G. Jay Kerns, Kjetil Halvorsen, William Asquith for their useful comments/suggestions. If you think information is not accurate or not complete, please send an e-mail to the maintainer or submit an issue or pull request in the GitHub repository linked above.

Table of contents

• Base functionality
• Discrete distributions
  • Univariate Discrete
  • Multivariate Discrete
• Continuous distributions
  • Univariate Continuous
  • Multivariate Continuous
• Other distributions
  • Mixed-type distributions
  • Mixture of probability laws
  • Random matrices
  • Copulas
  • Compound, composite, discretized, exponentiated and transformation of distributions
• Moments, skewness, kurtosis and etc
• Random number generators (RNG)
• Miscellaneous
• Bibliography

Base functionality

• Base R provides probability distribution functions pfoo() density functions dfoo(), quantile functions qfoo(), and random number generation rfoo() 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.
• Base R provides various one-sample or two-sample tests for univariate distributions, e.g., ks.test, shapiro.test, ansari.test, chisq.test, poisson.test. Ecume provides non-parametric two-sample (or k-sample) distribution comparisons in the univariate or multivariate case allowing observation weights and thresholds.
• Probability generating function: no longer implemented.

Some packages may optionally provide the symbolic derivatives with respect to the parameters for the probability functions. For instance, the first and second derivatives of the log-density can be of some help in estimation and inference tasks, and the derivatives of the quantile function can help when inferring on a given quantile. For that purpose, the following base R functions can be used stats::D() for derivatives w.r.t. a single parameter, or stats::deriv() for (partial) derivatives w.r.t. multiple parameters. The Deriv package provides a much more flexible symbolic differentiation interface. One can also use Stan Math library through StanHeaders package, see e.g. this blog. The nieve package provides symbolic differentiation for two probability distribution (Generalized Pareto and Generalized Extreme Value) in order to compute the log-likelihood for example.

Discrete distributions

Discrete univariate distributions

• Beta-binomial distribution: provided in VGAM, extraDistr, rmutil, emdbook. ZI/ZM beta binomial distributions are implemented in gamlss.dist.
• Beta-geometric distribution: provided in VGAM.

• Binomial (including Bernoulli) distribution: provided in stats. Zero-modified, zero-inflated, truncated versions are provided in gamlss.dist, extraDistr, actuar and in VGAM. LaplacesDemon provides dedicated functions for the Bernoulli distribution. rmutil provides the double binomial and the multiplicative binomial distributions.

  Summary for Binomial-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  binomial	stats	d, p, q, r	binom
  zero-infl. binomial	extraDistr	d, p, q, r	zib
  zero-infl. binomial	VGAM	d, p, q, r	zibinom
  zero-infl. binomial	gamlss.dist	d, p, q, r	ZIBI
  zero mod. binomial	VGAM	d, p, q, r	zabinom
  zero mod. binomial	actuar	d, p, q, r	zmbinom
  zero mod. binomial	gamlss.dist	d, p, q, r	ZABI
  zero trunc. binomial	actuar	d, p, q, r	ztbinom
  trunc. binomial	extraDistr	d, p, q, r	tbinom

• Bell Touchard distribution: standard and zero-inflated provided in countDM.
• Benford distribution: provided in VGAM and BenfordTests.
• Bernoulli distribution: provided in extraDistr.
• Borel-Tanner distribution: provided in VGAM.
• Delaporte distribution: provided in gamlss.dist and Delaporte.
• Dirac distribution: provided in distr.
• Discrete categorical distribution: provided in LaplacesDemon.
• Discrete exponential distribution: provided in poweRlaw.
• Discrete gamma distribution: provided in extraDistr.
• Discrete inverse Weibull distribution: DiscreteInverseWeibull provides d, p, q, r functions for the inverse Weibull as well as hazard rate function and moments.
• Discrete Laplace distribution: The discrete Laplace distribution is provided in extraDistr (d, p, r). The skew discrete Laplace distribution has two parametrization (DSL and ADSL), both provided in DiscreteLaplace and DSL in disclap. LaplacesDemon also provides the DSL parametrization only.
• Discrete lognormal distribution: provided in poweRlaw.
• Discrete normal distribution: provided in extraDistr.
• Discrete power law distribution: provided in poweRlaw.
• Discrete uniform distribution: can be easily obtained with the functions sum,cumsum,sample and is provided in extraDistr.
• Discrete Weibull distribution: provided in DiscreteWeibull: d, p, q, r, m for disc. Weib. type 1, d, p, q, r, m, h for disc. Weib. type 3. extraDistr provides d, p, q, r for Type 1.
• Felix distribution: provided in VGAM.
• gamma count distribution: provided in rmutil.
• Geometric distribution: provided in stats . Zero-modified, zero-inflated, truncated versions are provided in gamlss.dist, actuar and in VGAM. The time-varying geometric is provided in tvgeom.
• Geometric (compound) Poisson distribution (also known Polya-Aeppli distribution): provided in polyaAeppli.
• Generalized/fractional binomial distribution: GenBinomApps provides the generalized binomial distribution. frbinom provides the fractional binomial distribution where trials are from a generlized Bernoulli process.
• Generalized Hermite distribution: provided in hermite.
• Good distribution: provided in good.
• Hypergeometric distribution: provided in stats . Non-central hypergeometric distribution is provided in MCMCpack (d,r). Extended hypergeometric distribution can be found in BiasedUrn package, which provides not only p, d, q, r functions but also mean, variance, mode functions. Generalized hypergeometric distribution is implemented in SuppDists. Negative hypergeometric distribution is provided in tolerance, extraDistr.
• Lagrangian Poisson distribution: RMKdiscrete provides d, p, q, r functions for the univariate and the bivariate Lagrangian Poisson distribution.
• Lindley’s power series distribution: provided in LindleyPowerSeries.
• Logarithmic distribution: This can be found in extraDistr, VGAM, actuar, and gamlss.dist. Zero-modified and zero-truncated versions is provided in actuar. A fast random generator is available for the logarithmic distribution is implemented in Runuran as well as the ‘density’ function.
• Poisson distribution: provided in stats and in poweRlaw. Zero-modified, zero-inflated, truncated versions are provided in extraDistr, gamlss.dist, actuar and in VGAM. extraDistr provides the truncated Poisson distribution. LaplacesDemon provides the generalized Poisson distribution. rmutil provides the double Poisson, the multiplicative Poisson and the Power variance function Poisson distributions. poibin and PoissonBinomial provide the Poisson binomial distribution. See the mixture section such as the Poisson-lognormal mixture.
• Poisson-Lindley distribution: provided in tolerance.
• Power law distribution: provided in poweRlaw.
• Mana Clash distribution: provided in RMKdiscrete.
• Negative binomial distribution: provided in stats . Zero-modified, zero-inflated, truncated versions are provided in gamlss.dist, extraDistr, emdbook, actuar and in VGAM. New parametrization of the negative binomial distribution is available in RMKdiscrete. nbconv provides p, q, r functions for convolutions of negative binomial distributions.
• Sichel distribution: provided in gamlss.dist.
• Skellam distribution: provided in extraDistr, VGAM and skellam.
• Waring distribution: degreenet provides a random generator, cpd provides d, p, q, r functions for extended biparametric Waring.
• Yule-Simon distribution: provided in VGAM and sampling in degreenet.
• Zeta and Haight’s Zeta distribution: provided in VGAM, tolerance.

• Zipf distribution and extensions: d, p, q, r functions of the Zipf and the Zipf-Mandelbrot distributions are provided in tolerance, VGAM. Package zipfR provides tools for distribution of word frequency, such as the Zipf distribution. zipfextR provides three extensions of the Zipf distribution: the Marshall-Olkin Extended Zipf, the Zipf-Poisson Extreme and the Zipf-Poisson Stopped Sum distributions.

Discrete multivariate distributions

• Hyper Dirichlet distribution: provided in hyper2 package.
• Multinomial distribution: stats, mc2d, extraDistr packages provide d, r functions. r is provided in MultiRNG and compositions. p function is provided by pmultinom.
• Multinomial Dirichlet distribution: functions d, r are provided in MCMCpack, mc2d, dirmult, extraDistr and bayesm. r is provided in MultiRNG.
• Multivariate Ewens distribution: not yet implemented?
• Multivariate geometric: d, r functions provided in bivgeom for the bivariate geometric distribution. BivGeo provides the Basu-Dhar bivariate geometric distribution.
• Multivariate hypergeometric distribution: provided in extraDistr. The conditional hypergeometric distribution is provided (d, p, q, r) in chyper.
• Multivariate logarithmic distribution: the bivariate logarithmic distribution is provided in trawl.
  
• Multiplicative multinomial distribution: The multiplicative multinomial distribution is implemented in MM.
• Multivariate negative distribution: A bivariate distribution with negative-binomial marginals is available in RMKdiscrete and trawl. MNB provides a generator and diagnostic tool for multivariate negative binomial distribution.
  bzinb provides a random generator for the bivariate negative binomial (classic and zero-inflated) distribution.
• Multivariate Poisson distribution: compositions provides a random generator. bzinb provides a random generator for the bivariate Poisson (classic and zero-inflated) distribution.
• Multivariate Poisson-lognormal: the bivariate Poisson-lognormal distribution is provided in poilog.
• Multivariate Dirichlet (also known as Polya) distribution: functions d, r of the Dirichlet distribution are provided in extraDistr, LaplacesDemon, DirichletReg and Compositional.
• Truncated Stick-Breaking distribution: provided in LaplacesDemon.

Continuous distributions

Continuous univariate distributions

• Arcsine distribution: implemented in package distr.
• Argus distribution: implemented in package argus.

• Beta distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). extraDistr provides the beta distribution parametrized by the mean and the precision. actuar provides moments and limited expected values. sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central beta distribution for computing d, p, q, r functions. extraDistr provides the four-parameter beta with lower and upper bounds. The generalized beta of the first kind (GB1) (exponentiation of beta 1) is provided in gamlss.dist, mbbefd, actuar. betafunctions provides the four-parameter beta (that is with location and scale parameters), the beta parametrized by the mean and the variance as well as the beta compound beta distribution. The beta prime (or beta of the second kind), which is the distribution of X/(1-X) when X follows a beta distribution of the first kind, is provided in VGAM, extraDistr, LaplacesDemon and mc2d. The zero and one inflated beta distribution can be found in gamlss.dist. The generalized beta of the second kind (GB2) is provided in gamlss.dist, GB2. Several special cases of the generalized beta distribution are also implemented in VGAM, mc2d: Lomax, inverse Lomax, Dagum, Singh-Maddala, Pert distributions. actuar provides the Feller-Pareto distribution as special cases Burr, loglogistic, paralogistic, generalized Pareto, Pareto, see also the Pareto subsection. llogistic provides the log-logistic parametrized by the median.

  Summary for Beta-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Beta (1st kind)	stats	d, p, q, r	beta
  Beta	actuar	m, mgf, lev	beta
  Beta	betafunctions	d, p, q, r	Beta.4P
  Doubly non central beta	sadists	d, p, q, r	nbeta
  4-param beta	extraDistr	d, p, q, r	nsbeta
  zero-infl beta	gamlss.dist	d, p, q, r	BEZI
  one-infl beta	gamlss.dist	d, p, q, r	BEOI
  one-infl beta	mbbefd	d, p, q, r, m, ec	oibeta
  GB1	gamlss.dist	d, p, q, r	GB1
  GB1	mbbefd	d, p, q, r, m, ec	gbeta
  GB1	actuar	d, p, q, r, m, lev	genbeta
  one-infl GB1	mbbefd	d, p, q, r, m, ec	oigbeta

  Summary for Beta-2-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Beta (2nd kind)	VGAM	d, p, q, r	beta
  Beta (2nd kind)	extraDistr	d, p, q, r	invbeta
  Beta (2nd kind)	LaplacesDemon	d, r	betapr
  GB2	VGAM	d, p, q, r	genbetaII
  GB2	gamlss.dist	d, p, q, r	GB2
  GB2	GB2	d, p, q, r	gb2
  Trans beta 2	actuar	d, p, q, r, m, lev	trbeta

• Bell-G distribution: BGFD provides d, p, q, r functions for Bell exponential, Bell extended exponential, Bell Weibull, Bell extended Weibull, Bell-Fisk, Bell-Lomax, Bell Burr-XII, Bell Burr-X, complementary Bell exponential, complementary Bell extended exponential, complementary Bell Weibull, complementary Bell extended Weibull, complementary Bell-Fisk, complementary Bell-Lomax, complementary Bell Burr-XII and complementary Bell Burr-X distribution.
  The package also provides hazard function and an estimation procedure.
• Benini distribution: provided in VGAM.
• Bezier-Montenegro-Torres distribution: provided in BMT.
• Bhattacharjee (normal+uniform) distribution: provided in package extraDistr.
• Birnbaum-Saunders distribution: provided in bsgof, extraDistr, VGAM.
• Bridge distribution: provided in bridgedist, as detailed in Wang and Louis (2003). The distribution of random intercept that allows a marginalized random intercept logistic regression to also be logistic regression.
• Box Cox distribution: gamlss.dist provides the Box-Cox normal, the Box-Cox power exponential and the Box-Cox t distributions. rmutil provides the Box-Cox normal.
• Burr distribution: see Pareto.
• Cardioid distribution: provided in VGAM (d,p,q,r) and CircStats, circular (d,r).
• Carthwrite’s Power-of-Cosine distribution: provided in circular (d,r).
• Cauchy distribution: Base R provides the d, p, q, r functions for this distribution (see above). Other implementations are available in lmomco and sgt. The skew Cauchy distribution is provided in sn. LaplacesDemon provides d, p, q, r functions for the Half-Cauchy distribution. The wrapped Cauchy distribution is provided in CircStats.
• Chen distribution: no longer implemented.
• Chernoff distribution: ChernoffDist provides d, p, q functions of the distribution of the maximizer of the two-sided Brownian motion minus quadratic drift, known as Chernoff’s distribution.

• Chi(-squared or not) distribution: Base R provides the d, p, q, r functions for the chi-squared distribution, both central and non-central (see above). Moments, limited expected values and the moment generating function are provided in actuar. extraDistr provides d, p, q, r functions for inverse chi-squared distribution (standard and scaled). Only d,r functions are available for the inverse chi-squared distribution in package LaplacesDemon. A fast random generator is available for the Chi distribution is implemented in Runuran as well as the density function. The non-central Chi distribution is not yet implemented. The chi-bar-squared distribution is implemented in emdbook. sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for sums of non central chi-squared raised to powers distribution and sums of log of non central chi-squared for computing d, p, q, r functions.

  Summary for Chi-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Chi-squared	stats	d, p, q, r	chisq
  Chi-squared	actuar	m, mgf, lev	chisq
  Chi-squared	Runuran	d, r	chisq
  Chi-bar-squared	emdbook	d, p, q, r	chibarsq
  Chi	Runuran	d, r	chi
  Inverse Chi-squared	extraDistr	d, p, q, r	invchisq
  Scaled Inverse Chi-squared	extraDistr	d, p, q, r	invchisq
  Sum of power Chi-squared	sadists	d, p, q, r	sumchisqpow
  Sum of log Chi-squared	sadists	d, p, q, r	sumlogchisq

• Circular distribution: uniform circular provided in circular (d,r); Generalized von Mises circular provided in circular (d).
• Consul distribution: see rmutil.
• Continuous binomial distribution: cbinom provides the d/p/q/r functions for a continuous analog to the standard discrete binomial with continuous size parameter and continuous support with x in [0, size + 1].
• Dagum distribution: see beta.
• Davies distribution: The Davies distribution is provided in Davies package.
• (non-central) Dunnett’s test distribution: no longer provided.
• Eta-mu distribution: provided in lmomco. sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central eta distribution for computing d, p, q, r functions.

• 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 (or two-parameter exponential) and the truncated exponential distributions are implemented in lmomco and tolerance packages with d, p, q, r functions. Exponential Power distribution is also known as General Error Distribution: d, p, q, r functions for the power and the skew power exponential type 1-4 distributions are implemented in gamlss.dist and lmomco. The power exponential distribution is also provided in normalp, rmutil, LaplacesDemon. The skew power exponential is provided mixSPE. A fast random generator is available for the power Exponential distribution is implemented in Runuran as well as the density function. AEP implements the Asymmetric Exponential Power Distribution.

  Summary for exponential-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Exponential	stats	d, p, q, r	exp
  Exponential	actuar	m, mgf, lev	exp
  Exponential	gamlss.dist	d, p, q, r	EXP
  Exponential	poweRlaw	d, p, q, r	exp
  Inverse exponential	actuar	d, p, q, r, m, lev	invexp
  Shifted exponential	lmomco	d, p, q, r, lm, tlmr	exp
  Shifted exponential	tolerance	d, p, q, r	2exp
  Truncated exponential	lmomco	d, p, q, r, lm, tlmr	texp
  Truncated exponential	ReIns	d, p, q, r	texp
  Power exponential	normalp	d, p, q, r	normp
  Power exponential	Runuran	d, r	exp
  Power exponential	rmutil	d, r	powexp
  Power exponential	LaplacesDemon	d, p, q, r	pe
  Skew power exp.	lmomco	d, p, q, r, lm, tlmr	aep4
  Power and skew power exp.	mixSPE	r	pe, spe
  Power and skew power exp.	gamlss.dist	d, p, q, r	PE, SEP

• Externally studentized midrange distribution: Package SMR computes the studentized midrange distribution (d, p, q, r).
• Fisher-Snedecor (or F) distribution: Base R provides the d, p, q, r functions for the F distribution, possibly with a non-central parameter. sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central Fisher distribution (and product of multiple doubly non central Fisher distribution) for computing d, p, q, r functions. flexsurv provides d, p, q, r functions as well as hazard (h) and integrated hazard rate (i) functions for the generalized F distribution. fpow returns the noncentrality parameter of the noncentral F distribution if probability of type I and type II error, degrees of freedom of the numerator and the denominator are given.
• Frechet distribution: provided in VGAM, RTDE, ReIns, extraDistr, distributionsrd and evd. A fast random generator is available for the Frechet distribution is implemented in Runuran as well as the density function. The truncated Frechet distribution is provided in ReIns.
• Friedman’s Chi distribution: provided in SuppDists.

• Gamma distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). EnvStats provides d, p, q, r functions of the gamma parametrized by the mean and the coefficient of variation. actuar provides d, p, q, r functions of the inverse, the inverse transformed and the log gamma distributions while ghyp provides those functions for the variance gamma distribution. extraDistr and LaplacesDemon provide the inverse gamma distribution. CaDENCE provides the zero-inflated gamma distribution. VarianceGamma provides d, p, q, r functions for the variance gamma distribution as well as moments (skewness, kurtosis, ...). VGAM, ggamma provide d, p, q, r functions of the log gamma and the generalized gamma distribution. The generalized gamma distribution can also be found in gamlss.dist. See Pearson III for a three-parameter gamma distribution with a location parameter. flexsurv provides d, p, q, r functions as well as hazard (h) and integrated hazard rate (i) functions for the generalized gamma distribution. coga provides d, p, r functions for a sum of independent but not identically distributed gamma distributions. MCMCpack provides d, r functions of the Inverse Gamma. rmutil provides the generalized Gamma. distTails provides the full-tail gamma distribution sglg provides the generalized log-Gamma along with various functions to fit semi-parametric regression models. ollggamma provides d, p, q, r for the Odd Log-Logistic Generalized Gamma.

  Summary for gamma-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Gamma	stats	d, p, q, r	gamma
  Gamma	actuar	m, mgf, lev	gamma
  Gamma	EnvStats	d, p, q, r	gammaAlt
  zero-inflated Gamma	CaDENCE	d, p, q, r	bgamma
  Inverse gamma	actuar	d, p, q, r, m, lev, mgf	invgamma
  Inverse gamma	extraDistr	d, p, q, r	invgamma
  Inverse gamma	LaplacesDemon	d, r	invgamma
  Inverse gamma	MCMCpack	d, r	invgamma
  Log-gamma	actuar	d, p, q, r, m, lev	lgamma
  Log-gamma	VGAM	d, p, q, r	lgamma
  Variance gamma	ghyp	d, p, q, r	VG
  Variance gamma	VarianceGamma	d, p, q, r, m	vg
  Generalized gamma	flexsurv	d, p, q, r, h, i	gengamma
  Generalized gamma	gamlss.dist	d, p, q, r	GG
  Generalized gamma	VGAM	d, p, q, r	gengamma.stacy
  Generalized gamma	rmutil	d, p, q, r	ggamma
  Generalized gamma	ggamma	d, p, q, r	ggamma
  convolution of gamma	coga	d, p, r	coga
  Full-taill gamma	distTails	d, p, r	dFTG
  Generalized log-gamma	sglg	d, p, q, r	glg

• Pólya–Gamma distribution: r function random sampling routines for the distribution are provided by BayesLogit, pg, and pgdraw.

• 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. The truncnorm package provides d, p, q, r functions for the truncated gaussian distribution as well as functions for the first two moments. EnvStats provides d, p, q, r functions for the truncated normal distribution and the zero-modified distribution. extraDistr provides the truncated normal. LaplacesDemon provides d, p, q, r functions for the Half-normal distribution. The wrapped normal distribution is provided in CircStats. lmomco implements the generalized normal distribution. The Exponentially modified Gaussian is available in emg and gamlss.dist sn implements the skew normal distribution. greybox implements the folded normal distribution. VGAM implements the folded and the skewed normal distribution, and csn provides d, r functions for the closed skew normal distribution. NormalLaplace provides d, p, q, r functions for the sum of a normal and a Laplace random variables, while LaplacesDemon provides d, r functions of the sum of a normal and a Laplace random variables. PSDistr provides d, p, q, r functions of transformations of the normal distribution, such as expnormal and sinh-normal distributions.

  Summary for Gaussian-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Normal	stats	d, p, q, r	norm
  Normal	actuar	m, mgf	norm
  Truncated normal	truncnorm	d, p, q, r, m	truncnorm
  Truncated normal	EnvStats	d, p, q, r	normTrunc
  Truncated normal	extraDistr	d, p, q, r	tnorm
  Truncated normal	crch	d, p, q, r	cnorm
  Generalized normal	lmomco	d, p, q, r	gno
  Zero modified Gaussian	EnvStats	d, p, q, r	zmnorm
  Exponentially modified Gaussian	emg	d, p, q, r	emg
  Exponentially modified Gaussian	gamlss.dist	d, p, q, r	exGAUSS
  Folded and skew normal	gamlss.dist	d, p, q, r	SN1, SN2
  Folded normal	greybox	d, p, q, r	fnorm
  Closed skew normal	csn	d, p, q, r	csn
  Skew normal	sn	d, p, q, r	sn

• General error distribution (also known as exponential power distribution): see exponential item.
• Generalized extreme value distribution: d, p, q provided in lmomco; d, p, q, r, provided in VGAM, evd, evir, FAdist, extraDistr, EnvStats, TLMoments, rmutil, QRM, ROOPSD and fExtremes. revdbayes provide d,p,q,r functions of the GEV distribution in a Bayesian setting.
• Gompertz distribution: provided in flexsurv, extraDistr. flexsurv also provides hazard (h) and integrated hazard rate (i) functions. The shifted Gompertz distribution is implemented in extraDistr. The unit-Gompertz is provided in ugomquantreg.
• Govindarajulu distribution: provided in lmomco.
• Gumbel distribution: provided in packages lmomco, VGAM, gamlss.dist, FAdist, extraDistr, QRM, TLMoments, dgumbel, EnvStats and evd. actuar provides the raw moments and the moment generating function (mgf) in addition to the d, p, q, r functions. A fast random generator is available for the Gumbel distribution is implemented in Runuran as well as the density function. The reverse Gumbel distribution is implemented in lmomco and gamlss.dist. bgumbel provides the bimodel Gumbel distribution.
• Hjorth distribution: provided in rmutil.
• Huber distribution: Huber’s least favourable distribution provided in package smoothmest (d, r), and in VGAM, marg, extraDistr (d, p, q, r).
• (generalized) G-and-K, G-and-H distributions: gk provides d, p, q, r functions for the g-and-k and generalized g-and-h distributions which are nonlinear transforms of the Gaussian variables.
• (generalized) Hyperbolic distribution: fBasics, ghyp, GeneralizedHyperbolic and HyperbolicDist packages provide d, p, q, r functions for the generalized hyperbolic distribution. QRM provides d, r functions for the generalized hyperbolic distribution. SkewHyperbolic provides the skewed Hyperbolic Student t-Distribution. fBasics also implements the standardized generalized Hyperbolic distribution. A fast random generator is available for the hyperbolic distribution is implemented in Runuran as well as the density function.
• Hyperbolic sine distribution and extension: gamlss.dist provides the sinh and the asinh distributions. Generalized Power Hyperbolic sine distributions are provided in FatTailsR.
• Inverse Gaussian (also known Wald) distribution: d, p, q, and r functions of the inverse Gaussian are provided in statmod, extraDistr, SuppDists, rmutil. LaplacesDemon provides d, r functions for the inverse Gaussian distribution. actuar provides d, p, q, r, m, lev, mgf functions for the Inverse Gaussian distribution. SuppDists also provides a function that returns moments, skewness, kurtosis. fBasics the normal inverse Gaussian and standardized normal inverse Gaussian distributions. The generalized inverse gaussian (GIG) distribution can be found in gamlss.dist, ginormal, HyperbolicDist, QRM, rmutil. The truncated GIG is also available in ginormal. A random generator is available for the (generalized) Inverse Gaussian distribution is implemented in Runuran as well as the density function. GIGrvg generates random variables from the generalized inverse Gaussian distribution.
• Johnson distribution: provided in SuppDists. ForestFit provides d, p of Johnson SB distribution.
• Jones and Pewsey distribution: provided in circular (d).
• K-prime distribution: sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for K-prime distribution for computing d, p, q, r functions.
• Kappa distribution: A 4-parameter Kappa distribution is provided in lmomco and FAdist.
• Kappa-mu distribution: provided in lmomco.
• Kato-Jones distribution: provided in circular (d, r).
• Kendall’s tau distribution: provided in SuppDists.
• Kiener distribution: a family of distributions generalizing hyperbolic sine distributions (see hyperbolic sine section), d, p, q, r, m provided in FatTailsR.
• Kruskal Wallis distribution: provided in SuppDists.
• Kumaraswamy distribution: provided in packages VGAM, extraDistr and lmomco. elfDistr provides the Kumaraswamy Complementary Weibull Geometric Probability Distribution.
• (Tukey) Lambda distribution and its extensions: The generalized Lambda distribution (GLD) is well known for its wide range of shapes. The original Tukey Lambda distribution can be obtained as a special case of the generalized Lambda distribution. There exists different parametrization of GLD in the literature: RS (Ramberg-Schmeiser or tail-index param), FMKL (Freimer-Mudholkar-Kollia-Lin), FM5 (Five-parameter version of FKML by Gilchrist), GPD (gen. Pareto dist.) and AS (Asymmetry-steepness). The following packages implement such distributions (with d, p, q, r functions): gld (RS, FKML, FM5, GPD), Davies (RS), gb (RS), lmomco (FMKL), extraDistr (original Tukey). ecd provides the elliptic lambda distribution and its use for financial pricing.
• Tukey’s G/H distribution: no longer provided directly, but Tukey’s H distribution is provided as a special case of Lambert W x F distribution.
• Lambda-prime distribution: sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for K-prime distribution for computing d, p, q, r functions.
• Lambert W x F distribution: LambertW package provides d, p, q, r functions as well as the first 4 central moments and a qqplot.
• Laplace (also called double exponential distribution) and asymmetric Laplace distribution: provided in distr, lmomco, LaplacesDemon, L1pack, VGAM, sgt, extraDistr, greybox, rmutil and HyperbolicDist packages. LaplacesDemon provides the Laplace distribution parametrized by the precision parameter as well as the skew Laplace distribution. Asymetric Laplace distribution is implemented in ald, greybox. A fast random generator is available for the Laplace distribution is implemented in Runuran as well as the density function. smoothmest implements the density and the random generator. The skew Laplace distribution is available in sgt. LaplacesDemon provides the log-Laplace distribution.
• LASSO distribution: provided in LaplacesDemon.
• Lévy distribution: provided in rmutil.
• Lindley distribution: provided in VGAM and gambin.
• Linear failure rate distribution: no longer implemented.
• Loglog distribution: no longer implemented.
• Lomax distribution: see beta.

• Logistic distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). actuar and VGAM provide d, p, q, r functions for the log logistic (also called Fisk), the paralogistic and the inverse paralogistic distributions. FAdist the log-logistic distribution with two and three parameters. The generalized logistic distribution (Type I, also known as skew-logistic distribution) is provided in lmomco, sld, rmutil, SCI and glogis. GTDL implements generalized Time-Dependent Logistic distribution.

  Summary for Logistic-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Logistic	stats	d, p, q, r	logis
  Logistic	actuar	m, mgf	logis
  Log logistic	actuar	d, p, q, r, m, lev	llogis
  Log logistic	VGAM	d, p, q, r	fisk
  Log logistic	FAdist	d, p, q, r	llog, llog3
  Paralogistic	actuar	d, p, q, r, m, lev	paralogis
  Paralogistic	VGAM	d, p, q, r	paralogistic
  Inv. paralogistic	actuar	d, p, q, r, m, lev	invparalogis
  Inv. paralogistic	VGAM	d, p, q, r	inv.paralogistic
  Truncated logistic	crch	d, p, q, r	tlogis
  Generalized logistic	glogis	d, p, q, r	glogis
  Generalized logistic	SCI	d, p, q	genlog
  Generalized logistic	lmomco	d, p, q, r	glo
  Generalized logistic	sld	d, p, q, r	sl
  Generalized logistic	rmutil	d, p, q, r	glogis

• Logit-normal distribution: provided in logitnorm.
• Log-normal distribution and its extensions: The log normal distribution is implemented in Base R (see above) and poweRlaw. The log normal distribution parametrized by its mean and its coefficient of variation is also provided in EnvStats. LaplacesDemon provides the lognormal parametrized by the precision parameter. The truncated lognormal distribution is provided in EnvStats with two possible parametrizations as well as in ReIns. The 3-parameter lognormal distribution is available in lmomco, greybox, TLMoments, EnvStats and FAdist. The package loglognorm implements d, p, q, r functions for the double lognormal distribution, as well as the raw moment, the expected value and the variance functions. EnvStats provides d, p, q, r functions for the zero-modified lognormal distribution with two possible parametrizations. distributionsrd provides the double Pareto-lognormal distribution, the left Pareto-lognormal distribution, the truncated lognormal distribution.
• Makeham distribution: provided in VGAM and
• Maxwell distribution: provided in VGAM.
• Minimax distribution: provided in minimax.
• Mittag-Leffler distribution: d, p, q, r functions provided in MittagLeffleR.
• 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 lomax distribution) 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 I and II, the inverse Pareto, the ‘generalized pareto’ distributions, the Burr and the inverse Burr distributions, all special cases of the transformed beta II distribution. A fast random generator for the Burr and the Pareto II distribution is implemented in Runuran as well as the density. EnvStats and LaplacesDemon provides d, p, q, r functions for Pareto I distribution. extremefit provides the Burr, the Pareto II, mixture of Pareto I distributions and a composite distribution of two Pareto I distributions. lmomco, evd, fExtremes, extraDistr, QRM, Renext, revdbayes, FAdist, LaplacesDemon, TLMoments qrmtools and evir packages implement the Generalized Pareto Distribution (from Extreme Value Theory), which is depending the shape parameter’s value a Pareto II distribution, a shifted exponential distribution or a generalized beta I distribution. ParetoPosStable implements the Pareto positive stable distribution. The extended Pareto distribution is implemented in RTDE and the shifted truncated (to unit interval) Pareto is implemented in mbbefd. ReIns provides Burr, extended Pareto, generalized Pareto, Pareto 1 distributions and their truncated version. CaDENCE provides the Pareto 2 and the zero-inflated Pareto 2 distribution. Pareto provides the Pareto 1, piecewise Pareto and the generalized Pareto (from actuarial theory).

  Summary for Pareto-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Pareto I	VGAM	d, p, q, r	paretoI
  Pareto I	actuar	d, p, q, r, m, lev	pareto1
  Pareto I	EnvStats	d, p, q, r	pareto
  Pareto I	extraDistr	d, p, q, r	pareto
  Pareto I	ReIns	d, p, q, r	pareto
  Pareto I	LaplacesDemon	d, p, q, r	pareto
  Pareto I	distributionsrd	d, p, q, r	pareto
  Pareto I	Pareto	d, p, q, r	Pareto
  Trunc. Pareto I	ReIns	d, p, q, r	tpareto
  Pareto II	VGAM	d, p, q, r	paretoII
  Pareto II	actuar	d, p, q, r, m, lev	pareto, pareto2
  Pareto II	Runuran	d, r	pareto
  Pareto II	extraDistr	d, p, q, h	lomax
  Pareto II	extremefit	d, p, q, h	pareto
  Pareto II	Renext	d, p, q, r	lomax
  Pareto II	rmutil	d, p, q, r	pareto
  Pareto II	CaDENCE	d, p, q, r	pareto2
  zero-inflated Pareto II	CaDENCE	d, p, q, r	bpareto2
  Pareto III	VGAM	d, p, q, r	paretoIII
  Pareto III	actuar	d, p, q, r	pareto3
  Pareto IV	VGAM	d, p, q, r	paretoIV
  Pareto IV	actuar	d, p, q, r	pareto4
  Inverse Pareto	actuar	d, p, q, r, m, lev	invpareto
  Inverse Pareto	distributionsrd	d, p, q, r, m, lev	invpareto
  Extended Pareto	RTDE	d, p, q, r	EPD
  Extended Pareto	ReIns	d, p, q, r	epd
  Shift. trunc. Pareto	mbbefd	d, p, q, r, m, ec	stpareto
  Gen. Pareto (actuarial)	actuar	d, p, q, r, m, lev	genpareto
  Gen. Pareto (actuarial)	Pareto	d, p, q, r	GenPareto
  Gen. Pareto (EVT)	lmomco	d, p, q, r	gpa
  Gen. Pareto (EVT)	evd	d, p, q, r	gpd
  Gen. Pareto (EVT)	fExtremes	d, p, q, r	gpd
  Gen. Pareto (EVT)	evir	d, p, q, r	gpd
  Gen. Pareto (EVT)	extraDistr	d, p, q, r	gpd
  Gen. Pareto (EVT)	QRM	d, p, q, r	GPD
  Gen. Pareto (EVT)	ReIns	d, p, q, r	gpd
  Gen. Pareto (EVT)	LaplacesDemon	d, r	gpd
  Gen. Pareto (EVT)	TLMoments	d, p, q, r	gpd
  Trunc. Gen. Pareto (EVT)	ReIns	d, p, q, r	tgpd
  Gen. Pareto (EVT)	revdbayes	d, p, q, r	gp
  Gen. Pareto (EVT)	Renext	d, p, q, r	GPD
  Gen. Pareto (EVT)	qrmtools	d, p, q, r	GPD
  Gen. Pareto (EVT)	ROOPSD	d, p, q, r	gpd
  Feller-Pareto	actuar	d, p, q, r, m, lev	fpareto
  Burr	actuar	d, p, q, r, m, lev	burr
  Burr	extremefit	d, p, q, r	burr
  Burr	ReIns	d, p, q, r	burr
  Burr	rmutil	d, p, q, r	burr
  Trunc. Burr	ReIns	d, p, q, r	tburr
  Inverse Burr	actuar	d, p, q, r, m, lev	invburr

• Pearson’s distribution: Pearson type III available in lmomco and FAdist. A log-Pearson type III distribution is also available in FAdist. PearsonDS provides the d, p, q, r functions as well as the first four moments for the Pearson distributions: types I, II, III, IV, V, VI, VII. cpd provides d, p, q, r for complex bi/triparametric Pearson distributions.
• Pearson’s Rho distribution: provided in SuppDists.
• Perks distribution: provided in VGAM.
• Planck’s distribution: a random generator is available in Runuran.
• Phase-type distribution: provided in actuar, mapfit, matrixdist, PhaseTypeR.
• Power distribution: r pkg(“poweRlaw”)` implement the exponential power distribution. Two-sided power distribution provided in rmutil.
• Proportion distribution: this is the distribution for the difference between two independent beta distributions. d, p, q, r functions in tolerance.
• Quadratic forms and their ratios: CompQuadForm provides several exact and approximate methods to evaluate the distribution function of quadratic forms in normal variables. Qapprox provides fast approximations for the distribution function in nonnegative definite cases. QF provides d, p, q, r for nonnegative definite quadratic forms in normal variables and their ratios where the numerator and denominator are independent, as well as p for ratios of central quadratic forms in the same normal variables. qfratio provides d, p, q, r for the distribution of ratios of potentially noncentral quadratic forms in the same normal variables, as well as moment.
• Rayleigh distribution: provided in packages VGAM, extraDistr and lmomco.
• Response time distribution: rtdists provides d, p, q, r functions for the (Ratcliff) diffusion distribution and for the linear ballistic accumulator (LBA) with different underlying drift-distributions (Normal, Gamma, Frechet, and log-normal).
• Rice distribution: provided in VGAM and lmomco.
• Simplex distribution: provided in rmutil.
• Singh-Maddala distribution: see beta.
• Slash distribution: provided in lmomco, extraDistr and VGAM.
• Spearman’s Rho distribution: provided in SuppDists.
• Stable distribution: d, p, q, r functions are available in fBasics and stabledist, the functions use the approach of J.P. Nolan for general stable distributions. stable (d, p, q, r, h) is also used for general stable and uses a modified Buck parametrization. MixedTS provides mixed tempered stable distribution (d, p, q, r). FMStable provides (d, p, q) the extremal or maximally skew stable and the finite moment log stable distributions. SymTS provides d, p, q, r functions for symmetric stable, symmetric classical tempered stable, and symmetric power tempered stable distributions. TempStable provides d, p, q, r functions for tempered stable distributions. libstable4u provides d, p, q, r functions for skew stable distributions. dstabledist provides d, p, r functions for skew stable distributions.

• Student distribution and its extensions: Base R provides the d, p, q, r functions for Student and non central Student distribution (see above). extraDistr and LaplacesDemon provides the Student distribution with location and scale parameters. LaplacesDemon provides d, p, q, r functions for the Half-Student distribution. sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central Student distribution for computing d, p, q, r functions. The skewed Student distribution is provided in skewt, sn and gamlss.dist packages. The generalized skew distribution is provided in sgt. d, p, q, r functions for the generalized t-distribution can be found in gamlss.dist. fBasics provides d, p, q, r functions for the skew and the generalized hyperbolic t-distribution. The L-moments of the Student t (3-parameter) are provided in lmomco. crch provides d, p, q, r functions for the truncated student distribution.

  Summary for Student-related distributions

  Distribution name	Packages	Functions	Distribution suffix
  Student	stats	d, p, q, r	t
  Student with loc. and scal.	extraDistr	d, p, q, r	lst
  Student with loc. and scal.	LaplacesDemon	d, p, q, r	st
  Doubly non central St.	sadists	d, p, q, r	dnt
  Skew Student	skewt	d, p, q, r	skt
  Skew Student	sn	d, p, q, r	st
  Skew St. Type 1-5	gamlss.dist	d, p, q, r	ST1, ST2, ST3, ST4, ST5
  Gen. Student	gamlss.dist	d, p, q, r	GT
  Gen. Hyp. Student	fBasics	d, p, q, r	ght
  Skew Gen. Student	sgt	d, p, q, r	sgt

• Triangle/trapezoidal distribution: packages triangle, extraDistr, mc2d, EnvStats and VGAM provide d, p, q, r functions for the triangle or triangular distribution, while the package trapezoid provides d, p, q, r functions for the Generalized Trapezoidal Distribution. CircStats, circular provide d, r functions for triangular distribution. A fast random generator is available for the triangle distribution is implemented in Runuran as well as the density function.
• Tsallis or q-Exponential distribution: tsallisqexp provides d, p, q, r functions for two parametrizations of the Tsallis distribution and also implements a left-censored version.
• 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. KScorrect provides d, p, q, r functions for the log-uniform distribution.
• Upsilon distribution: sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for Upsilon distribution for computing d, p, q, r functions.
• Vasicek distribution: vasicek implements d, p, r functions. vasicekreg implements d, p, q, r functions.
  
• von Mises distribution: The CircStats package provides d, p, r functions; the circular package provides d, p, q, r functions. rvMF package provides a fast random generator for von Mises Fisher distribution.
• Wakeby distribution: A 5-parameter Wakeby is 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 in actuar package and also the moments and the limited expected value for both the raw and the inverse Weibull distribution. FAdist implements the three-parameter Weibull distribution. Furthermore, lmomco implements the Weibull distribution while evd implements the reverse Weibull distribution. The reverse generalized extreme value distribution are provided in gamlss.dist (d, p, q, r) and the shifted left truncated Weibull distribution is provided in Renext. The right truncated Weibull is provided in ReIns. The generalized Weibull is provided in rmutil. The tail Weibull is provided in distTails. CaDENCE provides the zero-inflated Weibull distribution.

• First-passage time of a Wiener process: WienR provides d, p functions of the first-passage time of a diffusion model.

Continuous multivariate distributions

• Bivariate Pareto: Bivariate.Pareto provides a random generator for the bivariate Pareto distribution.
• Multivariate beta distribution: NonNorMvtDist provides d, p, q, r, s functions for inverted beta distribution.
• Multivariate Burr distribution: NonNorMvtDist provides d, p, q, r, s functions.
• Multivariate Cauchy distribution: sn provide d, p, r functions for the multivariate skew Cauchy distribution, while LaplacesDemon provides d, r functions for the multivariate Cauchy distribution parametrized either by sigma, by the Cholesky decomposition of sigma, by the precision matrix omega or by the Cholesky decomposition of omega. mcauchyd provides d, r function of the multivariate Cauchy distribution.
• Cook-Johnson’s Multivariate Uniform Distribution: NonNorMvtDist provides d, p, q, r, s functions.
• Multivariate Dirichlet distribution: Compositional, LaplacesDemon, MCMCpack packages provide d, r functions as well as a fitting function for Compositional. compositions, bayesm provide r function. SGB provides a generalization of the Dirichlet distribution called Simplicial Generalized Beta distribution.
• Multivariate exponential distribution: while LaplacesDemon provides d, r functions for the multivariate power exponential distribution parametrized either by sigma, or by the Cholesky decomposition of sigma.
• Multivariate F distribution: NonNorMvtDist provides d, p, q, r, s functions.
• Multivariate Gaussian (or normal) distribution: The multivariate Gaussian distribution is provided in the packages mvtnorm (d, p, r), mnormt (d, p, r), mnorm (d, p, r), mniw (d, r), Compositional (r), compositions (r). pbv provides d, p functions for bivariate normal distributions. symmoments computes central and non-central moments of the multivariate Gaussian distribution. LaplacesDemon provides d, r functions for the multivariate normal distribution parametrized either by sigma, by the Cholesky decomposition of sigma, by the precision matrix omega or by the Cholesky decomposition of omega. Futhermore, the multivariate truncated normal is implemented in TruncatedNormal for d, p, r functions; tmvtnorm for p, q, r, m(oments) functions; tmvmixnorm for a fast RNG. sparseMVN implements very fast algorithms to compute the density and generate random variates of a multivariate normal distribution for which the covariance matrix or precision matrix is sparse. cmvnorm implements the complex multivariate normal distribution (d, r). Furthermore, condMVNorm implements d, p, r functions for the conditional multivariate normal distribution. condTruncMVN implements d, p, r functions of the conditional truncated multivariate normal distribution. Finally, sn besides providing facilities for their distribution functions, sn allows the creation of S4 objects which encapsulate these distributions and provide facilities for plotting, summary, marginalization, conditioning, affine transformations of these S4 objects. Compositional provides random generator for the multivariate normal distribution on the simplex and multivariate skew normal distribution on the simplex. A random generator of the multivariate normal is provided in MultiRNG. mggd provides d, r function of the multivariate generalized Gaussian distribution.
• Multivariate generalized hyperbolic distribution: QRM provides d, r functions of the standard and the symmetric multivariate generalized hyperbolic distribution. ghyp provides d, p, r functions of the standard multivariate generalized hyperbolic distribution.
• Multivariate generalized extreme value distribution: Both bivariate and multivariate Extreme Value distributions as well as order/maxima/minima distributions are implemented in evd (d, p, r).
• Multivariate Laplace distribution: LaplacesDemon provides d, r functions for the multivariate Laplace distribution parametrized either by sigma, or by the Cholesky decomposition of sigma. r is provided in MultiRNG. L1pack provides d, r functions of the multivariate Laplace distribution.
• Multivariate logistic distribution: VGAM package implements the bivariate logistic distribution, while NonNorMvtDist implements the multivariate logistic distribution.
• Multivariate lognormal distribution: compositions provides r function.
• Multivariate Pareto distribution: evd provides the density for the multivariate generalized Pareto type I. NonNorMvtDist provides d, p, q, r, s functions for multivariate Lomax (type II) distributions and its generalized version. NonNorMvtDist provides d, p, q, r, s functions for Mardia’s Multivariate Pareto Type I Distribution
• Multivariate Stable distribution: For elliptically contoured (subgaussian stable), alphastable provides d, r functions as well as a fitting function, mvgb provides p function. The multivariate subgaussian stable distribution (d, p, r) is available in mvpd.
• Multivariate Student distribution: The multivariate Student distribution is provided in the packages mvtnorm (d, r), mnormt (d, p, r), Compositional (r), tmvmixnorm (r), QRM (d, r), bayesm (r), MVT (r). TruncatedNormal for d, p, r functions; tmvtnorm for d, p, q, r functions. sn provides d, p, r functions for the multivariate skew t distribution. LaplacesDemon provides d, r functions for the multivariate Student distribution parametrized either by sigma, by the Cholesky decomposition of sigma, by the precision matrix omega or by the Cholesky decomposition of omega. Random generator r is provided in MultiRNG. A special case of a bivariate noncentral t-distribution called Owen distribution is provided in OwenQ.
• Multivariate Uniform distribution: r is provided in MultiRNG. compositions provides a random generator on the simplex.

Other distributions

Mixed-type distributions

• Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD) distribution : provided in mbbefd.
• Mixed ordinal and normal distribution: provided in OrdNor.
• One-inflated distributions: a generic distribution as well as special cases (OI-beta, OI-uniform, OI-GB1, OI-Pareto) are provided in mbbefd. The zero and one inflated beta distribution can be found in gamlss.dist.
• Zero-modified distributions: EnvStats provides the zero-modified normal distribution and the zero-modified lognormal distribution.

Mixture of probability laws

• Bernoulli-dist mixture: d, p, q, r functions for Bernoulli-exponential, Bernoulli-Gamma, Bernoulli-lognormal, Bernoulli-Weibull distributions are provided in qmap.
• Cauchy-polynomial quantile mixture: d, p, q, r functions are provided in Lmoments.
• Chi-square mixture: d, p, q, r functions are provided in emdbook.
• Gaussian mixture: Functions d, r are provided in mixtools, bmixture package when dealing with finite mixture models. nor1mix, extraDistr, mclust, LaplacesDemon, KScorrect provides d, p, r functions for Gaussian mixture. EnvStats provides d, p, q, r functions for mixture of two normal distributions. bayesm provides d function for the mixture of multivariate normals.
• Gamma Poisson: provided in extraDistr.
• Gamma mixture: Ga GSM package provides d, p, r, bmixture provides d, r, evmix provides d, p, q, r.
• Generic mixtures: there is an implementation via S4-class UnivarMixingDistribution in package distr. gendist provides d, p, q, r functions for two-distribution mixture models working with any distribution defined by its d, p, q, r functions.
• Horseshoe distribution: provided in LaplacesDemon.
• Laplace mixture distribution: provided in LaplacesDemon.
• Log normal mixture: d, p, q, r functions are provided in EnvStats with two possible parametrizations.
• Normal-polynomial quantile mixture: d, p, q, r functions are provided in Lmoments.
• Pareto distribution: extremefit implements the mixture of two Pareto I distributions.
• Poisson beta distribution: provided in scModels.
• Poisson Binomial distribution: poibin implements the Poisson Binomial distribution.
• Poisson lognormal distribution: poilog implements the Poisson lognormal distribution.
• Poisson mixture: provided in extraDistr.
• Poisson-Tweedie exponential family models: provided in poistweedie.
• Student mixture: The AdMit package provides d, r functions for Student mixtures in the context of Adaptive Mixture of Student-t distributions. bmixture package also provide d, r functions for mixture of Student-t distributions.
• von Mises Fisher (or Langevin) mixture: The movMF and CircStats packages provide d, r functions for finite von Mises Fisher mixtures.

Random matrices

• Huang-Wan distribution: provided in LaplacesDemon.
• Inverse matrix gamma distribution: provided in LaplacesDemon.
• Inverse Wishart distribution: LaplacesDemon provides inverse Wishart distribution parametrized either by Sigma or by its Cholesky decomposition. LaplacesDemon provides the scaled inverse Wishart distribution. MCMCpack and mniw provides the inverse Wishart distribution.
• Marcenko-Pastur distribution: provided in RMTstat, MCMCpack and bayesm.
• Matrix gamma distribution: provided in LaplacesDemon.

• Matrix normal distribution: MBSP (r) provides a random generator using a Cholesky decomposition; matrixsampling (r) provides a random generator using a spectral decomposition; LaplacesDemon and mniw (d, r); matrixNormal (d, p, r) collects these forms in one place and allows users to be flexible in simulating random variates (Cholesky, spectral, SVD).

• Matrix student distribution: provided in mniw.
• Normal Inverse Wishart distribution: provided in LaplacesDemon, mniw.
• Normal Wishart distribution: provided in LaplacesDemon.
• Tracy-Widom distribution: provided in RMTstat, MCMCpack and bayesm: supported beta values are 1 (Gaussian Orthogonal Ensemble), 2 (Gaussian Unitary Ensemble), and 4 (Gaussian Symplectic Ensemble).
• Sparse matrix: spam provides functionalities to draw random numbers from a user-supplied RNG (e.g. rexp) or from a multivariate normal distribution for large sparse matrices: typically for sparse covariance matrices.
• Spiked Wishart Maximum Eigenvalue Distribution: provided in RMTstat, MCMCpack and bayesm.
• Wishart distributions: Base R provides the r function for the Wishart distribution. MCMCpack, RMTstat, bayesm, mniw provides d, r functions, bayesm provides r function. LaplacesDemon provides Wishart distribution parametrized either by Sigma or by its Cholesky decomposition.
• White Wishart Maximum Eigenvalue Distribution: provided in RMTstat, MCMCpack and bayesm.
• Yang-Berger distribution: provided in LaplacesDemon.

• Zellner distribution: provided in LaplacesDemon.

Copulas

• Unified approaches: The packages fCopulae, copula, and copBasic provide a lot of general functionality for copulas. Although lacking support for many existing copulas themselves, copBasic is primarily oriented around utility functions for the general mathematics of copulas as described in the well known introduction to copulas by Nelsen.
• Archimedean copulas: gumbel is a standalone package for the Gumbel copula fCopulae implements the 22 Archimedean copulas of Nelsen (1998, Introduction to Copulas , Springer-Verlag) including Gumbel, Frank, Clayton, and Ali-Mikhail-Haq. VGAM provides Ali-Mikhail-Haq, Clayton, Frank, Frechet copulas. copula provides Ali-Mikhail-Haq, Clayton, Frank, Gumbel and Joe copulas. The Frank bivariate distribution is available in RTDE. VineCopula provides Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7 and BB8 copulas. Nested Archimedean copulas are available in the HAC package. copBasic provides functions for Ali-Mikhail-Haq, Clayton, Frechet copulas. QRM provides pdf and random generator for Clayton, Gumbel, Frank, BB9 copula. Bivariate.Pareto provides a random generator for the Frank copula with Pareto margins. nCopula, HAC provide hierarchical archimedean copulas. lcopula provides the Liouville copula. CopulaGAMM provides the bivariate version of Frank, FGM, Galambos, Gumbel, Huesler-Reiss, Joe, MTCJ, Plackett copulas.
• Blomqvist copula: provided in copBasic.
• Composition of copula: copBasic provides functions for composition of a single symmetric copula and composition of two copulas.
• Cubic copula: Not yet implemented?
• Dirichlet copula: Not yet implemented?
• Empirical copula: provided in copBasic, copent, HAC. GenOrd provides sampling function for multivariate discrete random vectors with a specified correlation matrix.
• Elliptical copulas: Gaussian, Student and Cauchy copulas are implemented in fCopulae for the bivariate cases. copula, VGAM, VineCopula provide the Gaussian and the Student copulas. QRM provides pdf and random generator for Gaussian, Student copulas. relliptical provides a random generator for multivariate truncated Normal, Student-t, Power Exponential, Pearson VII, Slash and Contaminated Normal distributions. CopulaGAMM provides the bivariate Gaussian and student copula.
• Extreme value copulas: fCopulae provides the following copulas Gumbel, Galambos, Husler-Reiss, Tawn, or BB5. copula implements Gumbel, Galambos and Husler-Reiss.
• Eyraud-Farlie-Gumbel-Morgenstern copula: provided in VGAM, RTDE, and copula.
• Integrated gamma copula: provided in igcop.
• Mardia copula: Not yet implemented?
• Nested copulas: arbitrary nested versions of copulas can be implemented in copula.
• Plackett: provided in VGAM, copBasic and copula.
• Vine copulas: Package vines provides functions for C- and D-vine copulas and VineCopula for general R-vine copulas.

Compound, composite, discretized, exponentiated and transformation of distributions

• Absolute value or half distribution: Half-Cauchy, half normal and half-student are implemented both in extraDistr and in LaplacesDemon.
• Composite distribution also known as spliced distribution: Split-normal (also known as the two-piece normal distribution) not yet implemented. Split-student provided in package dng. evmix provides d, p, q, r of the following composite distributions: gamma-GPD, lognormal GPD, normal-GPD, Weibull-GPD as well as bulk models such as GPD-normal-GPD distribution. gendist provides d, p, q, r functions for composite models working with any distribution defined by its d, p, q, r functions.
• Compound distribution: kdist provides d, p, q, r functions of the K distribution.
• Discretized distribution: distcrete allows discretised versions of continuous distribution by mapping continuous values to an underlying discrete grid, based on a (uniform) frequency of discretisation, a valid discretisation point, and an integration range.
• Quantile-based asymmetric (QBA) family of distributions: no longer implemented.
• Transformed distribution: Newdistns provides G-transformed distributions for a selected number of distributions which includes Marshall Olkin G distribution, exponentiated G distribution, beta G distribution, gamma G distribution, Kumaraswamy G distribution, generalized beta G distribution, beta extended G distribution, gamma G distribution, gamma uniform G distribution, beta exponential G distribution, Weibull G distribution, log gamma G1/G2 distribution, exponentiated generalized G distribution, exponentiated Kumaraswamy G distributions, geometric exponential Poisson G distribution, truncated-exponential skew-symmetric G distribution, modified beta G distribution, and exponentiated exponential Poisson G distribution. MPS provides also G-transformed distributions, such as beta exponential G distribution, beta G distribution, exponentiated exponential Poisson G distribution, exponentiated G distribution, exponentiated generalized G distribution, exponentiated Kumaraswamy G distribution, gamma uniform G distribution, gamma uniform type I/II G distribution, generalized beta G distribution, geometric exponential Poisson G distribution, gamma-X family of modified beta exponential G distribution, exponentiated exponential Poisson G distribution, gamma-X generated of log-logistic-X familiy of G distribution, Kumaraswamy G distribution, log gamma G type I/II distribution, modified beta G distribution, Marshall-Olkin Kumaraswamy G distribution, odd log-logistic G distribution, truncated-exponential skew-symmetric G distribution, T-X{log-logistic}G distribution, Weibull G distribution. gendist provides d, p, q, r functions for composite models, folded models, skewed symmetric models and arctan models working with any distribution defined by its d, p, q, r functions. ComRiskModel provides also G-transformed such as binomial-G, complementary negative binomial-G and complementary geometric-G families of distributions taking baseline models such as exponential, extended exponential, Weibull, extended Weibull, Fisk, Lomax, Burr-XII and Burr-X.
• Truncated distribution: A generic code snippet is available in the JSS . This code is now available in two packages: truncdist is a dedicated package providing d, p, q, r, m(oments) functions for a univariate truncated distribution given a user-supplied distribution; LaplacesDemon provides a generic function in a Bayesian environment.

Moments, skewness, kurtosis and etc

• Empirical mean, standard deviation and variance: base R provides mean(), sd(), var() functions to compute the mean, standard deviation and variance, respectively.
• Empirical skewness: available in agricolae, e1071, GLDEX, HyperbolicDist, modeest, moments, s20x, fromo, DistributionUtils, EnvStats, parameters packages.
• Empirical kurtosis: available in agricolae, DistributionUtils, e1071, EnvStats, GLDEX, HyperbolicDist, fromo, moments, parameters packages. The raw or centered moments are provided in e1071, moments.
• Empirical L-moments: L-moments are available in lmom, lmomco, Lmoments, GLDEX, EnvStats, trimmed L-moments are available in lmomco, TLMoments and Lmoments, right-censored L-moments are available in lmomco, and cumulants in GLDEX. TLMoments provides a function to convert them to some distribution parameters.
• Empirical probability weighted moments: Probability weighted moments are available in EnvStats and fromo.
• Empirical cumulants: fromo provides centered and standardized cumulants.
• Mode estimation: Package modeest provides mode computation of known distributions and mode estimation on datasets in the unimodal case. Package ModEstM provides mode estimation in unimodal and multimodal cases. Package multimode provides for testing and exploring the number of modes on data using non-parametric procedures.
• Order statistics: Distribution function of the jth order statistic can be obtained with base R functions. orders allows to generate samples of k-th order statistics and others quantities of interest for the following distributions: Burr, Feller-Pareto, Generalized Pareto, The Inverse Paralogistic, Marshall-Olkin G, exponentiated G, beta G, gamma G, Kumaraswamy G, generalized beta G, beta extended G, gamma G, gamma uniform G, beta exponential G, Weibull G, log gamma G I/II, exponentiated generalized G, exponentiated Kumaraswamy G, geometric exponential Poisson G, truncated-exponential skew-symmetric G, modified beta G, exponentiated exponential Poisson G, Poisson-inverse gaussian, Skew normal type 1, Skew student t, Sinh-Arcsinh, Sichel, Zero inflated Poisson.
• Empirical characteristic function: empichar evaluates the empirical characteristic function of univariate and multivariate samples.
• Dispersion index: Package GWI provides univariate dispersion index against a particular distribution.
  
• Theoretical moments:
  • common distributions: The actuar package implements raw moments, limited expected values and moment generating function for base R distributions. lmomco provides L-moments (L), trimmed L-moments (TL), and right-censored [RC] for the following distributions: Asymmetric Exponential Power (L), Cauchy (TL), Eta-Mu (L), Exponential (L), Gamma (L), Generalized Extreme Value (L), Generalized Lambda (L and TL), Generalized Logistic (L), Generalized Normal (L), Generalized Pareto (L[RC] and TL), Govindarajulu (L), Gumbel (L), Kappa (L), Kappa-Mu (L), Kumaraswamy (L), Laplace (L), Normal (L), 3-parameter log-Normal (L), Pearson Type III (L), Rayleigh (L), Reverse Gumbel (L[RC]), Rice/Rician (L), Slash (TL), 3-parameter Student T (L), Truncated Exponential (L), Wakeby (L), and Weibull (L). Multivariate L-moments (L-comoments).
  • hyperbolic distributions: HyperbolicDist provides the mean, variance, skewness, kurtosis, mode, raw and centered moments for the hyperbolic, the generalized hyperbolic and the generalized inverse Gaussian distributions.
  • Lambda distribution: GLDEX also provides the mean, variance, skewness, kurtosis of generalized Lambda distribution.
  • multivariate distributions: MomTrunc provides mean vector, covariance matrices and raw moments for truncated or folded of the following multivariate distributions: normal, skew normal, extended skew normal and student.

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). In addition to the tools above, setRNG provides an easy way to set, retain information about the setting, and reset the RNG.
• Pseudo-randomness: RDieHarder offers several dozen new RNGs from the GNU GSL. randtoolbox provides more recent RNGs such as SF Mersenne-Twister and WELL, which are generators of Mersenne Twister type, but with improved quality parameters. SuppDists implements two RNGs of G. Marsaglia. dqrng provides PCG family by O’Neill (2014) as well as Xoroshiro128+ and Xoshiro256+ by Blackman and Vigna (2018).
  • Support for several independent streams: rstream focuses on multiple independent streams of random numbers from different sources (in an object oriented approach). dqrng provides RNG for parallel computation either in R or in C++.
  • For non-uniform generation, the Runuran package interfaces to the UNU.RAN library for universal non-uniform generation as well as customised distributions based on polynomial interpolation of the inverse cumulative distribution function. rust performs non-uniform random variate generation from unimodal (low-dimensional) multivariate continuous distributions, using the generalized ratio-of-uniforms method. UnivRNG provides 17 non-uniform generators either using an acceptance/rejection algorithm or the inverse CDF method. MultiRNG provides 11 multivariate generators, see each distribution. Tinflex provides a non-uniform random number generator for quite arbitrary distributions with piecewise twice differentiable densities.
  • kernelboot provides functions for random generation from univariate and multivariate kernel densities (in particular multivariate Gaussian kernels).
• 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 and mc2d packages implement the latin hypercube sampling, an hybrid quasi/pseudo random method. sfsmisc also provides the Halton sequence. qrng provides Korobov, generalize Halton and Sobol quasi-random sequences. spacefillr provides Halton and Sobol sequences.
• 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.
• Parallel computing: Random-number generators for parallel computing are available via the rlecuyer package. See the HighPerformanceComputing task view for more details.
• Multivariate random vectors: for parametric multivariate distributions, we refer to Multivariate Continuous and Multivariate Discrete. For non-parametric distributions, SimJoint offers various to simulate multivariate distributions with non-parametric marginals given a Pearson or Spearman correlation matrix.
• Unit sphere and other: simdd provides a generator for the Fisher Bingham distribution on the unit sphere, the matrix Bingham distribution on a Grassmann manifold, the matrix Fisher distribution on SO(3), and the bivariate von Mises sin model on the torus. uniformly provides sampling on various geometric shapes, such as spheres, ellipsoids, simplices. watson allows simulating mixtures of Watson distributions.

Miscellaneous

• Computation/benchmark:
  • Approximation of d, p, q, r functions: PDQutils provides tools for computing the density, cumulative distribution, and quantile functions of a distribution when the cumulants or moments are given, using the classical Gram Charlier, Edgeworth and Cornish-Fisher approximations. sadists is a showcase for PDQutils, providing density, cumulative distribution, quantile, and random generation for the doubly non-central t, doubly non-central F, K-prime, Lambda-prime, Upsilon, and sum of (non-central) chi-squares to powers distributions. Various approximations and alternative computations for d, p, q functions of probability distributions in R are given DPQ.
  • benchden implements the 28 distributions introduced as kernel benchmarks for nonparametric density estimation by Berlinet and Devroye (1994): includes d, p, q, r functions as well as additional information on features of the densities.
  • For non-uniform generation, see the Runuran above.
• Non parametric models:
  • Binned Empirical distributions: The HistogramTools package provides a number of methods for manipulating empirical data that has been binned into histogram form, including: (1) the empirical cumulative distribution function, (2) the empirical quantile, and (3) information loss metrics associated with binning.
  • Empirical distribution: Base R provides functions for univariate analysis: (1) the empirical density (see density()), (2) the empirical cumulative distribution function (see ecdf()), (3) the empirical quantile (see quantile()) and (4) random sampling with or without replacement (see sample()). distributionsrd provides d, p, q, r user-friendly functions for the empirical distributions as well as moments. mded provides a function for measuring the difference between two independent or non-independent empirical distributions and returning a significance level of the difference. MEPDF provides functions to compute and visualize empirical density functions for multivariate data.
  • Non Parametric distributions : spd provides the Semi Parametric Piecewise Distribution, while fBasics implements spline smoothed distributions.
• 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.
• Unified interface to handle distributions:
  • S3 Object-orientation: distributions3 provides tools to create and to manipulate probability distributions using S3, that is distributions3, generics random(), pdf(), cdf() and quantile() provide replacements for base R’s r/d/p/q style functions. distributional also provides tools to create and to manipulate probability distributions using S3, with cdf(), density(), hdr(), mean(), median(), quantile(),...
  • S4 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, distrEllipse).
  • R6 Object-orientation: ROOPSD provides a R6 class interface to classic statistical distribution.
  • 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 above.
• Transversal functions:
  • Histogram, tail plots, distance estimation: DistributionUtils provides log-histogram, tail plots, functions for testing distributions using inversion tests and the Massart inequality. visualize provides functions to plot the pdf or pmf with highlights on area or when probability is present in user defined locations, as well as the graph is the mean and variance of the distribution. visualize provides lower tail, bounded, upper tail, and two tail calculations. visualize contains convenience functions for constructing and plotting bivariate probability distributions (probability mass functions, probability density functions and cumulative distribution functions). vistributions provides visualization tools for a selected number of distributions.
  • Parameter estimation: lmomco and Lmoments focus on univariate/multivariate (L-)moments estimation. VGAM provides a lot of parameter estimation for usual and “exotic” distributions. gaussDiff provides a collection difference measures for multivariate Gaussian probability density functions Package MASS implements the flexible fitdistr function for parameter estimations. fitdistrplus greatly enlarges fitdistr and enhances the tools to fit a user-supplied probability distribution. OneStep is based upon fitdistrplus to provide one-step estimation procedures. EnvStats, fitteR, ExtDist also provide tools to fit and select a set of probability distributions. flexsurv and msm provides a quantile function for a generic distribution based on numerical computation based on a dichotomic search.

Bibliography

General books

• N. L. Johnson, S. Kotz, N. Balakrishnan (1994). Continuous univariate distributions, Volume 1, Wiley.
• N. L. Johnson, S. Kotz, N. Balakrishnan (1995). Continuous univariate distributions, Volume 2, Wiley.
• N. L. Johnson, S. Kotz, N. Balakrishnan (1997). Discrete multivariate distributions, Wiley.
• N. L. Johnson, A. W. Kemp, S. Kotz (2008). Univariate discrete distributions, Wiley. doi:10.1002/0471715816
• S. Kotz, N. Balakrishnan, N. L. Johnson (2000). Continuous multivariate distributions Volume 1, Wiley.
• G. Wimmer (1999), Thesaurus of univariate discrete probability distributions.

Books dedicated to a distribution family

• M. Ahsanullah, B.M. Golam Kibria, M. Shakil (2014). Normal and Student’s t Distributions and Their Applications, Springer. doi:10.2991/978-94-6239-061-4
• B. C. Arnold (2010). Pareto Distributions, Chapman and Hall. doi:10.1201/b18141
• N. Balakrishnan (2014). Handbook of the Logistic Distribution, CRC Press. doi:10.1201/9781482277098

Books with applications

• C. Forbes, M. Evans, N. Hastings, B. Peacock (2011). Statistical Distributions, Wiley. doi:10.1002/9780470627242
• K. Krishnamoorthy (2015). Handbook of Statistical Distributions with Applications, Chapman and Hall. doi:10.1201/b19191
• Z. A. Karian, E. J. Dudewicz, K. Shimizu (2010). Handbook of Fitting Statistical Distributions with R, CRC Press. doi:10.1201/b10159-3

CRAN packages

Related links

• Clickable diagram of distribution relationships
• Compendium of distributions.
• Comprehensive list of data types
• Diagram of discrete distribution relationships
• Diagram of continuous distribution relationships
• Journal of Statistical Software: R programs for truncated distributions
• List and diagram of distribution relationship.

Other resources

• CRAN Task View: HighPerformanceComputing