The MCMC Procedure

Standard Distributions

The section Univariate Distributions (Table 54.7 through Table 54.34) lists all univariate distributions that PROC MCMC recognizes. The section Multivariate Distributions (Table 54.35 through Table 54.38) lists all multivariate distributions that PROC MCMC recognizes. With the exception of the multinomial distribution, all these distributions can be used in the MODEL, PRIOR, and HYPERPRIOR statements. The multinomial distribution is supported only in the MODEL statement. The RANDOM statement supports a limited number of distributions; see Table 54.4 for the complete list.

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 GENERAL and DGENERAL functions. See the section Specifying a New Distribution for more details. See the section Truncation and Censoring for tips about how to work with truncated distributions and censoring data.

Univariate Distributions

Table 54.7 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 $\text{[math]}$ , use the inversion method; if $\text{[math]}$ , use a uniform random number generator.

Table 54.8 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 54.9 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 54.10 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 54.11 $\text{[math]}$ 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 54.12 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 54.13 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 exponential distribution	$\text{[math]}$

Table 54.14 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 54.15 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 54.16 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 54.17 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 54.18 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 54.19 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 54.20 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 54.21 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 54.22 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 54.23 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 54.24 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 54.25 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 54.26 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 54.27 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 54.28 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 54.29 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 54.30 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 54.31 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 54.32 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 54.33 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 54.34 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.

Multivariate Distributions

Table 54.35 Dirichlet Distribution
PROC specification	$\text{[math]}$ dirich( $\text{[math]}$ ), where $\text{[math]}$ , for $\text{[math]}$
Density	$\text{[math]}$ , where $\text{[math]}$
Parameter restriction	$\text{[math]}$
Range	$\text{[math]}$ , $\text{[math]}$
Mean	$\text{[math]}$
Mode	$\text{[math]}$

Table 54.36 Inverse Wishart Distribution
PROC specification	$\text{[math]}$ iwishart( $\text{[math]}$ , $\text{[math]}$ ), both $\text{[math]}$ and $\text{[math]}$ are $\text{[math]}$ matrics
Density	$\text{[math]}$
Parameter restriction	$\text{[math]}$ must be symmetric and positive definte; $\text{[math]}$
Range	$\text{[math]}$ is symmetric and positive definite
Mean	$\text{[math]}$
Mode	$\text{[math]}$

Table 54.37 Multivariate Normal Distribution
PROC specification	$\text{[math]}$ mvn( $\text{[math]}$ , $\text{[math]}$ ), where $\text{[math]}$ , for $\text{[math]}$ , and $\text{[math]}$ is a $\text{[math]}$ variance matrix
Density	$\text{[math]}$
Parameter restriction	$\text{[math]}$ must be symmetric and positive definite
Range	$\text{[math]}$
Mean	$\text{[math]}$
Mode	$\text{[math]}$

Table 54.38 Multinomial Distribution
PROC specification	$\text{[math]}$ multinom( $\text{[math]}$ ), where $\text{[math]}$ and $\text{[math]}$ , for $\text{[math]}$
Density	$\text{[math]}$ , where $\text{[math]}$
Parameter restriction	$\text{[math]}$ with all $\text{[math]}$
Range	$\text{[math]}$ , nonnegative integers
Mean	$\text{[math]}$

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.