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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(, )

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

Table 52.4 Binary Distribution

PROC specification

binary()

density

parameter restriction

range

mean

round

variance

mode

random number

generate . If , ; else,

Table 52.5 Binomial Distribution

PROC specification

binomial(, )

density

parameter restriction

range

mean

variance

mode

Table 52.6 Cauchy Distribution

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)

Table 52.7 Distribution

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

Table 52.8 Exponential Distribution

PROC specification

expchisq()

density

parameter restriction

range

mode

random number

generate , and is a draw from the exponential distribution

relationship to the distribution

Table 52.9 Exponential Exponential 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

Table 52.10 Exponential Gamma 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

Table 52.11 Exponential Inverse 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

Table 52.12 Exponential Inverse-Gamma 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


Table 52.13 Exponential Scaled Inverse 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

Table 52.14 Exponential 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

Table 52.15 Gamma 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)

Table 52.16 Geometric Distribution

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

Table 52.17 Inverse Distribution

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


Table 52.18 Inverse-Gamma 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

Table 52.19 Laplace (Double Exponential) 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


Table 52.20 Logistic 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

Table 52.21 LogNormal 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


Table 52.22 Negative Binomial Distribution

PROC specification

negbin(, )

density

parameter restriction

range

mean

round

variance

mode

random number

generate , and (Fishman; 1996).

Table 52.23 Normal Distribution

PROC specification

normal(, sd = )

normal(, var = )

normal(, prec = )

density

parameter restriction

range

same

same

mean

same

same

variance

mode

same

same


Table 52.24 Pareto Distribution

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{}.

Table 52.25 Poisson Distribution

PROC specification

poisson()

density

parameter restriction

range

mean

variance

, if

mode

round

Table 52.26 Scaled Inverse Distribution

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.

Table 52.27 T 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.

Table 52.28 Uniform 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)


Table 52.29 Wald Distribution

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).

Table 52.30 Weibull Distribution

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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