generates a random sample from a multivariate Student’s distribution

The inputs are as follows:

N

is the number of desired observations sampled from the multivariate Student’s distribution.

DF

is a scalar value that represents 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 that contains random draws from the Student’s distribution with DF degrees of freedom, mean vector Mean, and covariance matrix Cov.

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).