Last updated on 20191017
by Christophe Dutang, Patrice Kiener
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.
Ultimate bibles on probability distributions are:
 different volumes of N. L. Johnson, S. Kotz and N. Balakrishnan books, e.g. Continuous Univariate Distributions, Vol. 1,
 Thesaurus of univariate discrete probability distributions by G. Wimmer and G. Altmann.
 Statistical Distributions by M. Evans, N. Hastings, B. Peacock.
 Distributional Analysis with Lmoment Statistics using the R Environment for Statistical Computing, Asquith (2011).
The maintainer gratefully acknowledges 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 let me know.

Base R provides probability distribution functions
p
foo()
density functions d
foo()
, quantile functions
q
foo()
, and random number generation
r
foo()
where foo indicates the type
of distribution:
beta (foo = beta
),
binomial binom
,
Cauchy cauchy
,
chisquared 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
.
 Probability generating function:
Compounding provides pgf for
xxx
distribution,
inverse xxx
distribution,
first derivative of the xxx
distribution,
where xxx
belongs to binomial, binomialPoisson, geometric, hypergeometric,
hyperPoisson, Katti type H1/H2, logarithmic, logarithmicbinomial,
logarithmicPoisson, negative binomial, Neyman type A/B/C, PascalPoisson, Poisson,
Poissonbinomial, PoissonLindley, PoissonPascal, Polya Aeppli, Thomas,
Waring, Yule.
 Betabinomial distribution:
provided in VGAM, extraDistr, rmutil,
emdbook.
ZI/ZM beta binomial distributions are
implemented in gamlss.dist.
 Betageometric distribution: provided in VGAM.
 Binomial (including Bernoulli) distribution: provided in stats.
Zeromodified, zeroinflated, 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 Binomialrelated distributions
Distribution name  Packages  Functions  Distribution suffix 
binomial  stats  d, p, q, r  binom 
zeroinfl. binomial  extraDistr  d, p, q, r  zib 
zeroinfl. binomial  VGAM  d, p, q, r  zibinom 
zeroinfl. 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 
 Benford distribution: provided in VGAM and BenfordTests.
 Bernoulli distribution: provided in extraDistr.
 BorelTanner distribution: provided in VGAM.
 Complex Pearson distribution: cpd provides
the complex biparamtric and triparametric Pearson distribution.
 ConwayMaxwellPoisson distribution: provided in
compoisson.
 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 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.
Zeromodified, zeroinflated, truncated versions are provided
in gamlss.dist, actuar and in VGAM.
 Geometric (compound) Poisson distribution
(also known PolyaAeppli distribution): provided in
polyaAeppli.
 Generalized binomial distribution: provided in
GenBinomApps.
 Generalized Hermite distribution: provided in
hermite.
 Hypergeometric distribution: provided in stats.
Noncentral 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.
Zeromodified and zerotruncated 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.
Zeromodified, zeroinflated, 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.
See the mixture section such as the Poissonlognormal mixture.
 PoissonLindley distribution: provided in tolerance.
 Power law distribution: provided in poweRlaw.
 Mana Clash distribution: provided in RMKdiscrete.
 Negative binomial distribution: provided in stats.
Zeromodified, zeroinflated, truncated versions are provided
in gamlss.dist, extraDistr, emdbook,
actuar and in VGAM.
New parametrization of the negative binomial distribution
is available in RMKdiscrete.
 Sichel distribution: provided in
gamlss.dist.
 Skellam distribution: provided in extraDistr,
VGAM and skellam.
 Waring distribution: sampling in degreenet.
 YuleSimon 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 ZipfMandelbrot 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 MarshallOlkin Extended Zipf,
the ZipfPoisson Extreme and
the ZipfPoisson Stopped Sum distributions.
 Bivariate binomial:
d, p functions provided in bivariate.
 Bivariate geometric:
d, r functions provided in bivgeom.
BivGeo provides the BasuDhar bivariate geometric distribution.
 Bivariate Poisson:
d, p functions provided in bivariate.
 Bivariate Poissonlognormal:
provided in poilog.
 Bivariate uniform:
d, p functions provided in bivariate.
 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.
 Negative multinomial distribution:
A bivariate distribution with negativebinomial marginals is available in
RMKdiscrete.
The multiplicative multinomial distribution is implemented in MM.
 Multivariate Poisson distribution:
compositions provides a random generator.
 Multivariate hypergeometric distribution:
provided in extraDistr.
 Multivariate Polya distribution:
functions d, r of the Dirichlet Multinomial (also known as multivariate Polya)
distribution are provided in extraDistr, LaplacesDemon
and Compositional.
 Multivariate Ewens distribution:
not yet implemented?
 Truncated StickBreaking distribution:
provided in LaplacesDemon.
 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).
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 CornishFisher approximations
for doubly non central beta distribution for computing d, p, q, r functions.
extraDistr provides the fourparameter 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.
The beta prime (or beta of the second kind), which is the distribution of X/(1X) 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, SinghMaddala, Pert distributions.
actuar provides the transformed beta 2 distribution which includes
as special cases Burr, loglogistic, paralogistic, generalized Pareto, Pareto,
see also the Pareto subsection.
llogistic provides the loglogistic parametrized by the median.
Summary for Betarelated distributions
Distribution name  Packages  Functions  Distribution suffix 
Beta (1st kind)  stats  d, p, q, r  beta 
Beta  actuar  m, mgf, lev  beta 
Beta  extraDistr  d, p, q, r  prop 
Doubly non central beta  sadists  d, p, q, r  nbeta 
4param beta  extraDistr  d, p, q, r  nsbeta 
zeroinfl beta  gamlss.dist  d, p, q, r  BEZI 
oneinfl beta  gamlss.dist  d, p, q, r  BEOI 
oneinfl 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 
oneinfl GB1  mbbefd  d, p, q, r, m, ec  oigbeta 
Summary for Beta2related 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 
 Benini distribution: provided in VGAM.
 BezierMontenegroTorres distribution: provided in BMT.
 Bhattacharjee (normal+uniform) distribution: provided in package
extraDistr.
 BirnbaumSaunders distribution: provided in package
VGAM and extraDistr.
 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 BoxCox normal,
the BoxCox power exponential and the BoxCox t distributions.
rmutil provides the BoxCox normal.
 Burr distribution: see Pareto.
 Cardioid distribution: provided in VGAM (d,p,q,r) and
CircStats, circular (d,r).
 Carthwrite's PowerofCosine 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 HalfCauchy
distribution.
The wrapped Cauchy distribution is provided in CircStats.
 Chen distribution: provided in reliaR.
 Chi(squared or not) distribution: Base R provides
the d, p, q, r functions for the chisquared distribution, both central
and noncentral (see above).
Moments, limited
expected values and the moment generating function are provided in actuar.
extraDistr provides d, p, q, r functions for inverse chisquared
distribution (standard and scaled).
Only d,r functions are available for the inverse chisquared distribution
in package geoR and LaplacesDemon.
A fast random generator is available for the Chi distribution
is implemented in Runuran as well as the density function.
The noncentral Chi distribution is not yet implemented. The chibarsquared distribution
is implemented in emdbook.
sadists implements Gram Charlier, Edgeworth and CornishFisher approximations
for sums of non central chisquared raised to powers distribution and sums of log of
non central chisquared for computing d, p, q, r functions.
Summary for Chirelated distributions
Distribution name  Packages  Functions  Distribution suffix 
Chisquared  stats  d, p, q, r  chisq 
Chisquared  actuar  m, mgf, lev  chisq 
Chisquared  Runuran  d, r  chisq 
Chibarsquared  emdbook  d, p, q, r  chibarsq 
Chi  Runuran  d, r  chi 
Inverse Chisquared  geoR  d, r  invchisq 
Inverse Chisquared  extraDistr  d, p, q, r  invchisq 
Scaled Inverse Chisquared  extraDistr  d, p, q, r  invchisq 
Sum of power Chisquared  sadists  d, p, q, r  sumchisqpow 
Sum of log Chisquared  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.
 (noncentral) Dunnett's test distribution: provided
in nCDunnett.
 Etamu distribution: provided in lmomco.
sadists implements Gram Charlier, Edgeworth and CornishFisher 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 twoparameter 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 14 distributions are implemented
in gamlss.dist and lmomco.
The power exponential distribution is also provided in normalp,
rmutil, LaplacesDemon and sgt.
The skew power exponential is provided sgt.
reliaR provides
the generalized exponential, the inverse generalized exponential,
the logistic exponential, the MarshallOlkin Extended Exponential and
the exponential extension distributions.
A fast random generator is available for the power Exponential distribution
is implemented in Runuran as well as the density function.
Summary for exponentialrelated 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 
Skew power exp.  lmomco  d, p, q, r, lm, tlmr  aep4 
Power and skew power exp.  gamlss.dist  d, p, q, r  PE, SEP 
Generalized and inverse gen. exp.  reliaR  d, p, q, r  gen.exp, inv.genexp 
Logistic, MarshallOlkin Ext. exp. and exp. ext.  reliaR  d, p, q, r  logis.exp, moee, exp.ext 
 Externally studentized midrange distribution:
Package SMR computes the studentized midrange distribution (d, p, q, r).
 FisherSnedecor (or F) distribution: Base R provides the d, p, q, r functions
for the F distribution, possibly with a noncentral parameter.
sadists implements Gram Charlier, Edgeworth and CornishFisher 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 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.
VarianceGamma provides d, p, q, r functions for the variance gamma distribution
as well as moments (skewness, kurtosis, ...).
VGAM provides 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. reliaR provides
the log gamma distribution. See Pearson III for a threeparameter
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 fulltail gamma distribution
Summary for gammarelated 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 
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 
Loggamma  actuar  d, p, q, r, m, lev  lgamma 
Loggamma  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 
convolution of gamma  coga  d, p, r  coga 
Fulltaill gamma  distTails  d, p, r  dFTG 
 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.
mvrtn provides random variates for left/right truncated normal distributions.
EnvStats provides d, p, q, r functions for the truncated normal distribution
and the zeromodified distribution.
extraDistr provides the truncated normal.
LaplacesDemon provides d, p, q, r functions for the Halfnormal
distribution.
The wrapped normal distribution is provided in CircStats.
lmomco implements the generalized normal distribution.
The Exponentially modified Gaussian is available in emg, gamlss.dist and
retimes.
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.
CompQuadForm provides the distribution function of quadratic forms
in normal variates.
NormalGamma provides the density of the sum of a gaussian and a gamma random variables.
NormalLaplace provides d, p, q, r functions for the sum of a normal
and a Laplace random variables, while LaplacesDemon provides
d, r function of the sum of a normal and a Laplace random variables.
Summary for Gaussianrelated 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  mvrtn  r, m  tn 
Truncated normal  EnvStats  d, p, q, r  normTrunc 
Truncated normal  extraDistr  d, p, q, r  tnorm 
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 
Exponentially modified Gaussian  retimes  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 and fExtremes.
evdbayes, revdbayes
provide d,p,q,r functions of the GEV distribution in a Bayesian setting.
 Gompertz distribution: provided in reliaR,
flexsurv, extraDistr. flexsurv also
provides hazard (h) and integrated hazard rate (i) functions.
The shifted Gompertz distribution is implemented in extraDistr.
 Govindarajulu distribution: provided in lmomco.
 Gumbel distribution: provided in packages lmomco,
VGAM, gamlss.dist, FAdist, extraDistr,
reliaR, QRM, TLMoments,
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.
 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) GandK, GandH distributions:
gk provides d, p, q, r functions for the gandk and generalized gandh
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 tDistribution.
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.
ihs provides the asinh distribution.
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 and STAR.
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 distribution can be found in gamlss.dist,
QRM, rmutil
and HyperbolicDist. 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.
frmqa computes p function of the generalized inverse Gaussian distribution.
 Johnson distribution: provided in SuppDists.
 Jones and Pewsey distribution: provided in circular (d).
 Kprime distribution: sadists implements Gram Charlier, Edgeworth and CornishFisher approximations
for Kprime distribution for computing d, p, q, r functions.
 Kappa distribution: A 4parameter Kappa distribution is provided in
lmomco and FAdist.
 Kappamu distribution: provided in lmomco.
 KatoJones 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.
 Kolmogorov distribution: p function provided in kolmim.
 Kruskal Wallis distribution: provided in SuppDists.
 Kumaraswamy distribution: provided in packages VGAM,
extraDistr and lmomco.
 (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 (RambergSchmeiser or tailindex param),
FMKL (FreimerMudholkarKolliaLin),
FM5 (Fiveparameter version of FKML by Gilchrist),
GPD (gen. Pareto dist.)
and AS (Asymmetrysteepness). 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 H distribution: provided as a special case of
Lambert W x F distribution.
 Lambdaprime distribution: sadists implements Gram Charlier, Edgeworth and CornishFisher approximations
for Kprime 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,
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 logLaplace distribution.
 LASSO distribution: provided in LaplacesDemon.
 Lévy distribution: provided in rmutil.
 Lindley distribution: provided in VGAM and gambin.
 Linear failure rate distribution: provided in reliaR.
 Loglog distribution: provided in reliaR
 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 loglogistic distribution with two and three parameters.
The generalized logistic distribution (Type I, also known as skewlogistic distribution)
is provided in lmomco, sld, rmutil,
SCI and glogis.
Summary for Logisticrelated 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 
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 
 Logitnormal distribution:
provided in logitnorm.
 Lognormal 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 3parameter 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 zeromodified lognormal
distribution with two possible parametrizations.
 Makeham distribution: provided in VGAM and
 Maxwell distribution: provided in VGAM.
 Minimax distribution: provided in minimax.
 MittagLeffler 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.
Summary for Paretorelated 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 
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 III  VGAM  d, p, q, r  paretoIII 
Pareto IV  VGAM  d, p, q, r  paretoIV 
Inverse Pareto  actuar  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 (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 
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 logPearson 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.
 Pearson's Rho distribution: provided in SuppDists.
 Perks distribution: provided in VGAM.
 Planck's distribution: a random generator is available in Runuran.
 Phasetype distribution: provided in actuar, see also
PhaseType for inference.
 Poisson subordinated distributions: provided in LIHNPSD
(d, p, q, r, m functions).
 Power distribution: reliaR and poweRlaw implement the
exponential power distribution.
Twosided 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.
 Rayleigh distribution: provided in packages VGAM, extraDistr
and lmomco. Generalized and logistic Rayleigh distributions are available in
reliaR.
 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
driftdistributions (Normal, Gamma, Frechet, and lognormal).
 Rice distribution: provided in VGAM and lmomco.
 Simplex distribution: provided in rmutil.
 SinghMaddala 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. 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.
 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 HalfStudent
distribution.
sadists implements Gram Charlier, Edgeworth and CornishFisher 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 tdistribution
can be found in gamlss.dist. fBasics provides d, p, q, r functions
for the skew and the generalized hyperbolic tdistribution.
The Lmoments of the Student t (3parameter) are provided in lmomco.
Summary for Studentrelated 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 15  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 qExponential distribution:
tsallisqexp provides d, p, q, r functions for two parametrizations of
the Tsallis distribution and also implements a leftcensored 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.
HI generates
uniformly random points on a bounded convex set, in particular the unit ball.
KScorrect provides d, p, q, r functions for the loguniform distribution.
 Upsilon distribution: sadists implements Gram Charlier, Edgeworth and CornishFisher approximations
for Upsilon distribution for computing 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.
 Wakeby distribution: A 5parameter 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 threeparameter Weibull distribution, while
reliaR implements the exponential Weibull, the flexible Weibull,
the generalized power weibull, the MarshallOlkin Extended Weibull and
the Weibull extension distributions.
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.
 Bivariate normal:
d, p functions provided in bivariate.
 Bivariate Pareto:
Bivariate.Pareto provides a random generator for the bivariate Pareto distribution.
 Bivariate uniform:
d, p functions provided in bivariate.
 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.
 Dirichlet distribution:
Compositional, LaplacesDemon, MCMCpack packages
provide d, r functions as well as a fitting function for Compositional.
compositions, bayesm provide r function.
bivariate provides d function for the trivariate Dirichlet.
 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 Gaussian (or normal) distribution:
The multivariate Gaussian distribution is provided in the packages mvtnorm (d, p, r),
mnormt (d, p, r), Compositional (r), compositions (r).
pbv provides d, p functions for bivariate normal distributions.
mvprpb computes the orthant probability of the multivariate Gaussian distribution.
symmoments computes central and noncentral 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).
Finally, condMVNorm implements d, p, r functions for the conditional multivariate
normal distribution.
Furthermore, 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.
mnormpow computes the expected product of the components of a multivariate Gaussian vector.
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.
 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.
 Multivariate logistic distribution:
VGAM package implements the bivariate logistic distribution.
 Multivariate lognormal distribution:
compositions provides r function.
 Multivariate Pareto distribution:
mgpd provides the density for the multivariate generalized Pareto distribution
of type II, while evd provides the density for type I.
 Multivariate Stable distribution:
not yet implemented?
 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).
The multivariate truncated student is implemented in
TTmoment for r (sampling) and moments (m);
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 tdistribution called Owen distribution
is provided in OwenQ.
 Multivariate Uniform distribution:
r is provided in MultiRNG.
compositions provides a random generator
on the simplex.
 MaxwellBoltzmannBoseEinsteinFermiDirac (MBBEFD) distribution :
provided in mbbefd.
 Mixed ordinal and normal distribution:
provided in OrdNor.
 Oneinflated distributions:
a generic distribution as well as special cases (OIbeta, OIuniform, OIGB1, OIPareto)
are provided in mbbefd.
The zero and one inflated beta distribution can be found in gamlss.dist.
 Zeromodified distributions:
EnvStats provides the zeromodified normal distribution
and the zeromodified lognormal distribution.
 Bernoullidist mixture: d, p, q, r functions
for Bernoulliexponential, BernoulliGamma, Bernoullilognormal,
BernoulliWeibull distributions are provided in qmap.
 Cauchypolynomial quantile mixture: d, p, q, r functions are
provided in Lmoments.
 Chisquare 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 S4class
UnivarMixingDistribution in package distr. gamlss.mx
uses the gamlss.dist package.
gendist provides d, p, q, r functions for twodistribution 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.
 Normalpolynomial quantile mixture: d, p, q, r functions are
provided in Lmoments.
 Pareto distribution: extremefit implements
the mixture of two Pareto I distributions.
 Poisson Binomial distribution: poibin implements
the Poisson Binomial distribution.
 Poisson lognormal distribution: poilog implements
the Poisson lognormal distribution.
 Poisson mixture: provided in extraDistr.
 PoissonTweedie 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 Studentt distributions.
MitISEM, bmixture package also provide d, r functions for mixture of Studentt distributions.
 von Mises Fisher (or Langevin) mixture:
The movMF and CircStats packages provide d, r functions
for finite von Mises Fisher mixtures.
 Absolute value or half distribution:
HalfCauchy, half normal and halfstudent are implemented
both in extraDistr and in LaplacesDemon.
 Composite distribution also known as spliced distribution:
Composite lognormal distributions provided in CompLognormal.
Splitnormal (also known as the twopiece normal distribution) not yet implemented.
Splitstudent provided in package dng.
evmix provides d, p, q, r of the following composite distributions:
gammaGPD, lognormal GPD, normalGPD, WeibullGPD
as well as bulk models such as GPDnormalGPD 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: d, p, q, r, m functions are implemented by Compounding
where the parent distribution is any continuous distribution and the compound distribution
is any distribution among the list: binomial, binomialPoisson, geometric, hypergeometric,
hyperPoisson, Katti type H1/H2, logarithmic, logarithmicbinomial,
logarithmicPoisson, negative binomial, Neyman type A/B/C, PascalPoisson, Poisson,
Poissonbinomial, PoissonLindley, PoissonPascal, Polya Aeppli, Thomas,
Waring, Yule.
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.
 transformed distribution: Newdistns provides Gtransformed 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, truncatedexponential skewsymmetric G distribution, modified beta
G distribution, and exponentiated exponential Poisson 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.
 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 usersupplied
distribution;
LaplacesDemon provides a generic function in a Bayesian environment.
 Wrapped G distribution:
Wrapped provides d, p, q, r functions for a large family of distributions.
 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, TSA, s20x, fromo,
DistributionUtils, EnvStats,
parameters packages.
 Empirical kurtosis:
available in agricolae, DistributionUtils,
e1071, EnvStats,
GLDEX, HyperbolicDist, fromo,
moments, TSA,
parameters packages.
The raw or centered moments are provided in e1071,
moments.
 Empirical Lmoments:
Lmoments are available in lmom, lmomco, Lmoments,
GLDEX, EnvStats,
trimmed Lmoments are available in lmomco, TLMoments
and Lmoments,
rightcensored Lmoments 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 estimation for various distributions.
 Order statistics:
Distribution function of the jth order statistic can be obtained with base R functions.
ORDER2PARENT transforms distribution function of order statistics to its parent
distribution function.
 Empirical characteristic function:
empichar evaluates the empirical characteristic function of
univariate and multivariate samples.
 Theoretical moments:
 common distributions:
The actuar package implements raw moments, limited expected values
and moment generating function for base R distributions.
lmomco provides Lmoments (L), trimmed Lmoments (TL),
and rightcensored [RC] for
the following distributions: Asymmetric Exponential Power (L), Cauchy (TL), EtaMu (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), KappaMu (L), Kumaraswamy (L), Laplace (L),
Normal (L), 3parameter logNormal (L), Pearson Type III (L), Rayleigh (L), Reverse Gumbel
(L[RC]), Rice/Rician (L), Slash (TL), 3parameter Student T (L), Truncated Exponential
(L), Wakeby (L), and Weibull (L). Multivariate Lmoments (Lcomoments).
 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.
 Normal distribution:
mvrtn provides mean, variance for left/right truncated normal distributions.
 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.
 HuangWan 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 provides the inverse Wishart distribution.
 MarcenkoPastur 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 (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).
 Normal Inverse Wishart distribution:
provided in LaplacesDemon.
 Normal Wishart distribution:
provided in LaplacesDemon.
 TracyWidom 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 usersupplied 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 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.
 YangBerger distribution:
provided in LaplacesDemon.
 Zellner distribution:
provided in LaplacesDemon.
 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, SpringerVerlag) including
Gumbel, Frank, Clayton, and AliMikhailHaq.
VGAM provides AliMikhailHaq, Clayton, Frank, Frechet copulas.
copula provides AliMikhailHaq, Clayton, Frank, Gumbel and Joe copulas.
The Frank bivariate distribution is available in RTDE.
CDVine and VineCopula provide
Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7 and BB8 copulas.
Nested Archimedean copulas are available in the HAC package.
Generalized Archimedean copulas are implemented in the fgac package.
BivarP provides cdf, pdf and survival function for Clayton, Gumbel and Frank copula.
copBasic provides functions for AliMikhailHaq, 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 provides hierarchical archimedean copulas.
lcopula provides the Liouville copula.
 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 and in 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, CDVine, VGAM,
VineCopula provide the Gaussian and the Student copulas.
QRM provides pdf and random generator for Gaussian, Student copulas.
 Extreme value copulas: fCopulae provides the following copulas
Gumbel, Galambos, HuslerReiss, Tawn, or BB5. copula implements
Gumbel, Galambos and HuslerReiss.
 EyraudFarlieGumbelMorgenstern copula: provided in VGAM, RTDE,
and copula.
 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: Packages CDVine, vines
provide functions for C and Dvine copulas and
VineCopula for general Rvine copulas.
 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 MersenneTwister algorithm.
Other generators include WichmannHill, MarsagliaMulticarry, SuperDuper,
KnuthTAOCP, KnuthTAOCP2002, as well as usersupplied RNGs. For normal
random numbers, the following algorithms are available: KindermanRamage,
AhrensDieter, BoxMuller, Inversion (default). In addition to the tools above, setRNG
provides an easy way to set, retain information about the setting, and reset the RNG.
 Pseudorandomness:
RDieHarder offers several dozen new RNGs
from the GNU GSL. randtoolbox provides
more recent RNGs such as SF MersenneTwister and WELL,
which are generators of Mersenne Twister type, but with
improved quality parameters.
rngwell19937 provides one of the WELL generators with 53 bit
resolution of the output and allows seeding by a vector
of integers of arbitrary length.
randaes provides the deterministic part of the Fortuna cryptographic
pseudorandom number generator (AES).
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).
The qqid package contains functions to retrieve true, quantumrandom QQIDs,
to generate pseudorandom QQID.
 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 nonuniform generation, the Runuran
package interfaces to the UNU.RAN library for universal nonuniform generation
as well as customised distributions based on polynomial interpolation of the inverse
cumulative distribution function.
rust performs nonuniform random variate generation from unimodal
(lowdimensional) multivariate continuous distributions, using the
generalized ratioofuniforms method.
UnivRNG provides 17 nonuniform generators either using an
acceptance/rejection algorithm or the inverse CDF method.
MultiRNG provides 11 multivariate generators, see each distribution.
 kernelboot provides functions for random generation from univariate
and multivariate kernel densities (in particular multivariate Gaussian kernels).
 Quasirandomness: 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 quasirandom sequences.
 True randomness: The random
package provides several functions that access the
true random number service at
random.org.
The qrandom provides the ANU Quantum Random Number Generator provided
by the Australian National University (based on measuring the quantum fluctuations
of the vacuum).
 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:
Randomnumber generators for parallel computing are available via
the rlecuyer package.
See the HighPerformanceComputing task view for more details.
 Computation:
 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 CornishFisher approximations.
sadists is a showcase for PDQutils, providing density, cumulative distribution,
quantile, and random generation for the doubly noncentral t, doubly noncentral F,
Kprime, Lambdaprime, Upsilon, and sum of (noncentral) chisquares to powers
distributions.
 For nonuniform generation, see the Runuran above.
 Benchmark: A set of 28 densities suitable for comparing
nonparametric density estimators in simulation studies can be found
in the benchden package. The densities vary greatly in degree of
smoothness, number of modes and other properties. The package
provides d,p,q and r functions.
 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 (see sample()).
mded provides a function for measuring the difference between two
independent or nonindependent empirical distributions and returning a
significance level of the difference.
MEPDF, Emcdf provide functions to compute and
visualize empirical density functions for multivariate data.
probhat computes nonparametric probability distributions (d, p, q) using
kernel smoothing. probhat supports univariate, multivariate and conditional
distributions, and weighted 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 Objectorientation:
distributions3 provides tools to create and to manipulate
probability distributions using S3.
With distributions3, generics
random()
, pdf()
,
cdf()
and quantile()
provide replacements for base R's r/d/p/q
style functions.
 S4 Objectorientation:
General discrete and continuous distributions are implemented in package
distr respectively via S4class 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 distrfamily (distrSim, distrTEst,
distrTeach, distrDoc, distrEllipse).
 R6 Objectorientation:
distr6 provides a complete R6 Probability Distributions Interface
for 42 probability distributions and 11 kernels including functionality for
multiple scientific types.
Additionally, distr6 gives some functionalities for composite
distributions and numerical imputation.
 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 Objectorientation above.
 User Interface:
AtelieR package provides a GTK GUI for teaching basic
concepts in statistical inference, implementing all the R base distributions
as well as the generalized Student, the inverse Chisquare, the inverse gamma
and the lambdaprime distributions.
 Transversal functions:
 Histogram, tail plots, distance estimation:
DistributionUtils provides loghistogram,
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 and enhances
the tools to fit any probability distribution.
EnvStats and fitteR also provides tools to fit most common distributions.
flexsurv and msm provides a quantile
function for a generic distribution based on numerical
computation based on a dichotomic search.