RANDWISHART Function :: SAS/IML(R) 13.1 User's Guide

RANDWISHART Function

RANDWISHART (N, DF, Sigma ) ;

The RANDWISHART function is part of the IMLMLIB library. The RANDWISHART function returns an $N \times (p \times p)$ matrix that contains $N$ random draws from the Wishart distribution with DF degrees of freedom. Each row of the returned matrix represents a $p\times p$ matrix.

The inputs are as follows:

N: is the number of desired observations sampled from the distribution.
DF: is a scalar value that represents the degrees of freedom, $\mbox{DF} \geq p$ .
Sigma: is a $p \times p$ symmetric positive definite matrix.

The Wishart distribution is a multivariate generalization of the gamma distribution. (Note, however, that Kotz, Balakrishnan, and Johnson (2000) suggest that the term “multivariate gamma distribution” should be restricted to those distributions for which the marginal distributions are univariate gamma. This is not the case with the Wishart distribution.) A Wishart distribution is a probability distribution for nonnegative definite matrix-valued random variables. These distributions are often used to estimate covariance matrices.

If a $p \times p$ nonnegative definite matrix $X$ follows a Wishart distribution with parameters $\nu$ degrees of freedom and a $p \times p$ symmetric positive definite matrix $\Sigma$ , then

the probability density function for $x$ is

$f(x; \nu , \Sigma ) = \frac{ |x|^{(\nu -p-1)/2} \exp (-\frac{1}{2}\, \mbox{trace}(x \, \Sigma ^{-1})) }{ 2^{p \nu /2} |\Sigma |^{\nu /2} \pi ^{p(p-1)/4} \prod _{i=1}^ p \Gamma (\frac{\nu -i+1}{2})}$
if $p=1$ and $\Sigma =1$ , then the Wishart distribution reduces to a chi-square distribution with $\nu$ degrees of freedom.
the expected value of $X$ is $\nu \Sigma$ .

The following example generates 1,000 samples from a Wishart distribution with 7 degrees of freedom and $2\times 2$ matrix parameter S. Each row of the returned matrix x represents a $2\times 2$ nonnegative definite matrix. (You can reshape the $i$ th row of x with the SHAPE function.) The example computes the sample mean and compares it with the expected value.

call randseed(1);
N=1000;
DF = 7;
S = {1 1, 1 5};
x = RandWishart( N, DF, S );
Mean = DF * S;
SampleMean = shape( x[:,], 2, 2);
print SampleMean Mean;

Figure 24.303: Estimated Mean of Matrices

SampleMean		Mean
7.0518633	7.2402925	7	7
7.2402925	36.056848	7	35

For further details about sampling from the Wishart distribution, see Johnson (1987).