PROC MCMC: Standard Distributions :: SAS/STAT(R) 9.22 User's Guide

The MCMC Procedure

Standard Distributions

Table 52.3 through Table 52.30 show all densities that PROC MCMC recognizes. These densities can be used in the MODEL, PRIOR, and HYPERPRIOR statements. See the section Using Density Functions in the Programming Statements for information about how to use distributions in the programming statements. To specify an arbitrary distribution, you can use the functions GENERAL and DGENERAL. See the section Specifying a New Distribution for more details. See the section Truncation and Censoring for tips on how to work with truncated distributions and censoring data.

Table 52.3 Beta Distribution
PROC specification	beta( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$ , $\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$
random number	if $\text{[math]}$ , see (Cheng; 1978); if $\text{[math]}$ , see (Atkinson and Whittaker; 1976) and (Atkinson; 1979); if $\text{[math]}$ and $\text{[math]}$ , see (Cheng; 1978); if $\text{[math]}$ or $\text{[math]}$ , inversion method; if $\text{[math]}$ , uniform random variable

Table 52.4 Binary Distribution
PROC specification	binary( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	round $\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$
random number	generate $\text{[math]}$ . If $\text{[math]}$ , $\text{[math]}$ ; else, $\text{[math]}$

Table 52.5 Binomial Distribution
PROC specification	binomial( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$

Table 52.6 Cauchy Distribution
PROC specification	cauchy( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	does not exist
variance	does not exist
mode	$\text{[math]}$
random number	generate $\text{[math]}$ , let $\text{[math]}$ . Repeat the procedure until $\text{[math]}$ . $\text{[math]}$ is a draw from the standard Cauchy, and $\text{[math]}$ (Ripley; 1987)

Table 52.7 $\text{[math]}$ Distribution
PROC specification	chisq( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$ if $\text{[math]}$ ; $\text{[math]}$ otherwise
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$ if $\text{[math]}$ ; does not exist otherwise
random number	$\text{[math]}$ is a special case of the gamma distribution: $\text{[math]}$ is a draw from the $\text{[math]}$ distribution

Table 52.8 Exponential $\text{[math]}$ Distribution
PROC specification	expchisq( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mode	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the exponential $\text{[math]}$ distribution
relationship to the $\text{[math]}$ distribution	$\text{[math]}$

Table 52.9 Exponential Exponential Distribution
PROC specification	expexpon(scale = $\text{[math]}$ )	expexpon(iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same
mode	$\text{[math]}$	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the exponential exponential distribution. Note that an exponential exponential distribution is not the same as the double exponential distribution.
relationship to the Expon distribution	$\text{[math]}$

Table 52.10 Exponential Gamma Distribution
PROC specification	expgamma( $\text{[math]}$ , scale = $\text{[math]}$ )	expgamma( $\text{[math]}$ , iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same
mode	$\text{[math]}$	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the exponential gamma distribution
relationship to the $\text{[math]}$ distribution	$\text{[math]}$

Table 52.11 Exponential Inverse $\text{[math]}$ Distribution
PROC specification	expichisq( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mode	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the exponential inverse $\text{[math]}$ distribution
relationship to the $\text{[math]}$ distribution	$\text{[math]}$

Table 52.12 Exponential Inverse-Gamma Distribution
PROC specification	expigamma( $\text{[math]}$ , scale = $\text{[math]}$ )	expigamma( $\text{[math]}$ , iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same
mode	$\text{[math]}$	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the exponential inverse-gamma distribution
relationship to the $\text{[math]}$ distribution	$\text{[math]}$

Table 52.13 Exponential Scaled Inverse $\text{[math]}$ Distribution
PROC specification	expsichisq( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mode	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the exponential scaled inverse $\text{[math]}$ distribution
relationship to the $\text{[math]}$ distribution	$\text{[math]}$

Table 52.14 Exponential Distribution
PROC specification	expon(scale = $\text{[math]}$ )	expon(iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same
mean	$\text{[math]}$	$\text{[math]}$
variance	$\text{[math]}$	$\text{[math]}$
mode	$\text{[math]}$	$\text{[math]}$
random number	the exponential distribution is a special case of the gamma distribution: $\text{[math]}$ is a draw from the exponential distribution

Table 52.15 Gamma Distribution
PROC specification	gamma( $\text{[math]}$ , scale = $\text{[math]}$ )	gamma( $\text{[math]}$ , iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$ if $\text{[math]}$ otherwise	same
mean	$\text{[math]}$	$\text{[math]}$
variance	$\text{[math]}$	$\text{[math]}$
mode	$\text{[math]}$ if $\text{[math]}$	$\text{[math]}$ if $\text{[math]}$
random number	see (McGrath and Irving; 1973)

Table 52.16 Geometric Distribution
PROC specification	geo( $\text{[math]}$ )
density ¹	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	round( $\text{[math]}$ )
variance	$\text{[math]}$
mode	$\text{[math]}$
random number	based on samples obtained from a Bernoulli distribution with probability $\text{[math]}$ until the first success

Table 52.17 Inverse $\text{[math]}$ Distribution
PROC specification	ichisq( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$ if $\text{[math]}$
variance	$\text{[math]}$ if $\text{[math]}$
mode	$\text{[math]}$
random number	inverse $\text{[math]}$ is a special case of the inverse-gamma distribution: $\text{[math]}$ is a draw from the inverse $\text{[math]}$ distribution

Table 52.18 Inverse-Gamma Distribution
PROC specification	igamma( $\text{[math]}$ , scale = $\text{[math]}$ )	igamma( $\text{[math]}$ , iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same
mean	$\text{[math]}$ if $\text{[math]}$	$\text{[math]}$ if $\text{[math]}$
variance	$\text{[math]}$	$\text{[math]}$
mode	$\text{[math]}$	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the $\text{[math]}$ distribution
relationship to the gamma distribution	$\text{[math]}$

Table 52.19 Laplace (Double Exponential) Distribution
PROC specification	laplace( $\text{[math]}$ , scale = $\text{[math]}$ )	laplace( $\text{[math]}$ , iscale = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same
mean	$\text{[math]}$	$\text{[math]}$
variance	$\text{[math]}$	$\text{[math]}$
mode	$\text{[math]}$	$\text{[math]}$
random number	inverse CDF. $\text{[math]}$ Generate $\text{[math]}$ . If $\text{[math]}$ else $\text{[math]}$ . $\text{[math]}$ is a draw from the Laplace distribution

Table 52.20 Logistic Distribution
PROC specification	logistic( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$
random number	inverse CDF method with $\text{[math]}$ . Generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the logistic distribution

Table 52.21 LogNormal Distribution
PROC specification	lognormal( $\text{[math]}$ , sd = $\text{[math]}$ )	lognormal( $\text{[math]}$ , var = $\text{[math]}$ )	lognormal( $\text{[math]}$ , prec = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same	same
mean	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
variance	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
mode	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the lognormal distribution

Table 52.22 Negative Binomial Distribution
PROC specification	negbin( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	round $\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$
random number	generate $\text{[math]}$ , and $\text{[math]}$ (Fishman; 1996).

Table 52.23 Normal Distribution
PROC specification	normal( $\text{[math]}$ , sd = $\text{[math]}$ )	normal( $\text{[math]}$ , var = $\text{[math]}$ )	normal( $\text{[math]}$ , prec = $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
parameter restriction	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
range	$\text{[math]}$	same	same
mean	$\text{[math]}$	same	same
variance	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
mode	$\text{[math]}$	same	same

Table 52.24 Pareto Distribution
PROC specification	pareto( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$ if $\text{[math]}$
variance	$\text{[math]}$ if $\text{[math]}$
mode	$\text{[math]}$
random number	inverse CDF method with $\text{[math]}$ . Generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the Pareto distribution.
useful transformation	$\text{[math]}$ is Beta( $\text{[math]}$ , 1)I{ $\text{[math]}$ }.

Table 52.25 Poisson Distribution
PROC specification	poisson( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$
variance	$\text{[math]}$ , if $\text{[math]}$
mode	round $\text{[math]}$

Table 52.26 Scaled Inverse $\text{[math]}$ Distribution
PROC specification	sichisq( $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$ if $\text{[math]}$
variance	$\text{[math]}$ if $\text{[math]}$
mode	$\text{[math]}$
random number	scaled inverse $\text{[math]}$ is a special case of the inverse-gamma distribution: $\text{[math]}$ is a draw from the scaled inverse $\text{[math]}$ distribution.

Table 52.27 T Distribution
PROC specification	t( $\text{[math]}$ , sd = $\text{[math]}$ , $\text{[math]}$ )	t( $\text{[math]}$ , var = $\text{[math]}$ , $\text{[math]}$ )	t( $\text{[math]}$ , prec = $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
parm restriction	$\text{[math]}$ , $\text{[math]}$	$\text{[math]}$ , $\text{[math]}$	$\text{[math]}$ , $\text{[math]}$
range	$\text{[math]}$	same	same
mean	$\text{[math]}$ if $\text{[math]}$	same	same
variance	$\text{[math]}$ if $\text{[math]}$	$\text{[math]}$ if $\text{[math]}$	$\text{[math]}$ if $\text{[math]}$
mode	$\text{[math]}$	same	same
random number	$\text{[math]}$ is a draw from the t-distribution.

Table 52.28 Uniform Distribution
PROC specification	uniform( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	none
range	$\text{[math]}$
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	does not exist
random number	Mersenne Twister (Matsumoto and Kurita; 1992, 1994; Matsumoto and Nishimura; 1998)

Table 52.29 Wald Distribution
PROC specification	wald( $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$
random number	generate $\text{[math]}$ . Let $\text{[math]}$ and $\text{[math]}$ . Perform a Bernoulli trial, $\text{[math]}$ . If $\text{[math]}$ , choose $\text{[math]}$ ; otherwise, choose $\text{[math]}$ (Michael, Schucany, and Haas; 1976).

Table 52.30 Weibull Distribution
PROC specification	weibull( $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ )
density	$\text{[math]}$
parameter restriction	$\text{[math]}$
range	$\text{[math]}$ if $\text{[math]}$ otherwise
mean	$\text{[math]}$
variance	$\text{[math]}$
mode	$\text{[math]}$ if $\text{[math]}$
random number	inverse CDF method with $\text{[math]}$ . Generate $\text{[math]}$ , and $\text{[math]}$ is a draw from the Weibull distribution.

Footnotes

The random variable $\text{[math]}$ is the total number of failures in an experiment before the first success. This density function is not to be confused with another popular formulation, $\text{[math]}$ , which counts the total number of trials until the first success.

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