Module Library |

**generates a random sample from a multivariate
Student's distribution**

**RANDMVT(***N, DF, Mean, Cov***)**

The inputs are as follows:

- is the number of desired observations sampled from the
multivariate Student's distribution.
*DF*- is a scalar value representing the degrees of freedom for the distribution.
*Mean*- is a vector of means.
*Cov*- is a symmetric positive definite variance-covariance matrix.

The RANDMVT function returns an matrix containing random draws from the Student's distribution with

If follows a multivariate distribution with degrees of freedom, mean vector , and variance-covariance matrix , then

- the probability density function for is
- if , the probability density function reduces to a univariate Student's distribution.
- the expected value of is .
- the covariance of and is when .

The following example generates 1000 samples from a two-dimensional distribution with 7 degrees of freedom, mean vector , and covariance matrix S. Each row of the returned matrix x is a row vector sampled from the distribution. The example then computes the sample mean and covariance and compares them with the expected values. Here are the code and the output:

call randseed(1); N=1000; DF = 4; Mean = {1 2}; S = {1 1, 1 5}; x = RandMVT( N, DF, Mean, S ); SampleMean = x[:,]; n = nrow(x); y = x - repeat( SampleMean, n ); SampleCov = y`*y / (n-1); Cov = (DF/(DF-2)) * S; print SampleMean Mean, SampleCov Cov; SampleMean Mean 1.0768636 2.0893911 1 2 SampleCov Cov 1.8067811 1.8413406 2 2 1.8413406 9.7900638 2 10

In the preceding example, the columns (marginals) of x do
*not* follow univariate distributions. If you want a sample whose
marginals are univariate , then you need to scale each column of
the output matrix:

x = RandMVT( N, DF, Mean, S ); StdX = x / sqrt(diag(S)); /* StdX columns are univariate t */

Equivalently, you can generate samples whose marginals are univariate by passing in a correlation matrix instead of a general covariance matrix.

For further details about sampling from the multivariate distribution, see Kotz and Nadarajah (2004, pp. 1 - 11).

Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.