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.

 PROC specification beta(, ) density parameter restriction , range mean variance mode random number if , see (Cheng; 1978); if , see (Atkinson and Whittaker; 1976) and (Atkinson; 1979); if and , see (Cheng; 1978); if or , inversion method; if , uniform random variable

 PROC specification binary() density parameter restriction range mean round variance mode random number generate . If , ; else,

 PROC specification binomial(, ) density parameter restriction range mean variance mode

 PROC specification cauchy(, ) density parameter restriction range mean does not exist variance does not exist mode random number generate , let . Repeat the procedure until . is a draw from the standard Cauchy, and (Ripley; 1987)

 PROC specification chisq() density parameter restriction range if ; otherwise mean variance mode if ; does not exist otherwise random number is a special case of the gamma distribution: is a draw from the distribution

 PROC specification expchisq() density parameter restriction range mode random number generate , and is a draw from the exponential distribution relationship to the distribution

 PROC specification expexpon(scale = ) expexpon(iscale = ) density parameter restriction range same mode random number generate , and 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

 PROC specification expgamma(, scale = ) expgamma(, iscale = ) density parameter restriction range same mode random number generate , and is a draw from the exponential gamma distribution relationship to the distribution

 PROC specification expichisq() density parameter restriction range mode random number generate , and is a draw from the exponential inverse distribution relationship to the distribution

 PROC specification expigamma(, scale = ) expigamma(, iscale = ) density parameter restriction range same mode random number generate , and is a draw from the exponential inverse-gamma distribution relationship to the distribution

 PROC specification expsichisq(, ) density parameter restriction range mode random number generate , and is a draw from the exponential scaled inverse distribution relationship to the distribution

 PROC specification expon(scale = ) expon(iscale = ) density parameter restriction range same mean variance mode random number the exponential distribution is a special case of the gamma distribution: is a draw from the exponential distribution

 PROC specification gamma(, scale = ) gamma(, iscale = ) density parameter restriction range if otherwise same mean variance mode if if random number see (McGrath and Irving; 1973)

 PROC specification geo() density 1 parameter restriction range mean round() variance mode random number based on samples obtained from a Bernoulli distribution with probability until the first success

 PROC specification ichisq() density parameter restriction range mean if variance if mode random number inverse is a special case of the inverse-gamma distribution: is a draw from the inverse distribution

 PROC specification igamma(, scale = ) igamma(, iscale = ) density parameter restriction range same mean if if variance mode random number generate , and is a draw from the distribution relationship to the gamma distribution

 PROC specification laplace(, scale = ) laplace(, iscale = ) density parameter restriction range same mean variance mode random number inverse CDF. Generate . If else . is a draw from the Laplace distribution

 PROC specification logistic(, ) density parameter restriction range mean variance mode random number inverse CDF method with . Generate , and is a draw from the logistic distribution

 PROC specification lognormal(, sd = ) lognormal(, var = ) lognormal(, prec = ) density parameter restriction range same same mean variance mode random number generate , and is a draw from the lognormal distribution

 PROC specification negbin(, ) density parameter restriction range mean round variance mode random number generate , and (Fishman; 1996).

 PROC specification normal(, sd = ) normal(, var = ) normal(, prec = ) density parameter restriction range same same mean same same variance mode same same

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

 PROC specification poisson() density parameter restriction range mean variance , if mode round

 PROC specification sichisq() density parameter restriction range mean if variance if mode random number scaled inverse is a special case of the inverse-gamma distribution: is a draw from the scaled inverse distribution.

 PROC specification t(, sd = , ) t(, var = , ) t(, prec = , ) density parm restriction , , , range same same mean if same same variance if if if mode same same random number is a draw from the t-distribution.

 PROC specification uniform(, ) density parameter restriction none range mean variance mode does not exist random number Mersenne Twister (Matsumoto and Kurita; 1992, 1994; Matsumoto and Nishimura; 1998)

 PROC specification wald(, ) density parameter restriction range mean variance mode random number generate . Let and . Perform a Bernoulli trial, . If , choose ; otherwise, choose (Michael, Schucany, and Haas; 1976).

 PROC specification weibull(, , ) density parameter restriction range if otherwise mean variance mode if random number inverse CDF method with . Generate , and is a draw from the Weibull distribution.
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