Module Library: RANDMVT Function :: SAS/IML(R) 9.2 User's Guide

Module Library

RANDMVT Function

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 DF degrees of freedom, mean vector Mean, and covariance matrix Cov.

If

follows a multivariate

distribution with $\nu$ degrees of freedom, mean vector $\mu$ , and variance-covariance matrix $\sigma$ , then

the probability density function for is
$f(x; \nu, \mu, \sigma) = \frac{\gamma((\nu+p)/2)}{|\sigma |^{1/2} (\pi \nu )^{p/2}\gamma(\nu/2)} ( 1+ \frac{(x-\mu) \sigma^{-1} (x-\mu)^t}{\nu} )^{-(\nu+p)/2}$
if , the probability density function reduces to a univariate Student's distribution.
the expected value of is $\mu_i$ .
the covariance of and is $\frac{\nu}{\nu-2}\sigma_{ij}$ when $\nu\gt 2$ .

The following example generates 1000 samples from a two-dimensional distribution with 7 degrees of freedom, mean vector $(1 \, 2)$ , 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).

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