## RANDWISHART Function

**generates a random sample from a Wishart distribution**

**RANDWISHART(** *N, DF, Sigma ***)**

The inputs are as follows:

- is the number of desired observations sampled from the distribution.

*DF*
- is a scalar value representing the degrees of
freedom, .
*Sigma*
- is a symmetric positive definite
matrix.

The RANDWISHART function returns an

matrix containing

random draws from the Wishart distribution with

*DF* degrees
of freedom. Each row of the returned matrix represents a

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

nonnegative definite matrix

follows a Wishart
distribution with parameters

degrees of freedom and a

symmetric positive definite matrix

, then

- the probability density function for is

- if and , then the Wishart distribution
reduces to a chi-square distribution
with degrees of freedom.
- the expected value of is .

The following example generates 1000 samples from a Wishart
distribution with 7 degrees of freedom and matrix
parameter S. Each row of the returned matrix x represents
a nonnegative definite matrix. (You can reshape the th
row of x with the SHAPE function.) The example then computes
the sample mean and compares them with the expected value. Here are
the code and the output:

call randseed(1);
N=1000;
DF = 7;
S = {1 1, 1 5};
x = RandWishart( N, DF, S );
ExpectedValue = DF * S;
SampleMean = shape( x[:,], 2, 2);
print SampleMean ExpectedValue;
SampleMean ExpectedValue
7.0518633 14.103727 7 14
14.103727 28.207453 14 28

For further details about sampling from the Wishart distribution, see
Johnson (1987, pp. 203 - 204).

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