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A multivariate Student’s t-distribution is derived by analogy to the derivation of a multivariate normal (Gaussian) probability density function. This multivariate Student’s t-distribution can have different shape parameters for the marginal probability density functions of the multivariate distribution. Expressions for the probability density function, for the variances, and for the covariances of the multivariate t-distribution with arbitrary shape parameters for the marginals are given.

An expression for a multivariate Student’s t-distribution is presented. This expression, which is different in form than the form that is commonly used, allows the shape parameter

The form that is typically used is [

This “typical” form attempts to generalize the univariate Student’s t-distribution and is valid when the n marginal distributions have the same shape parameter

The multivariate Student’s t-distribution put forth here is derived from a Cholesky decomposition of the scale matrix by analogy to the multivariate normal (Gaussian) pdf. The derivation of the multivariate normal pdf is given in Section 2 to provide background. The multivariate Student’s t-distribution and the variances and covariances for the multivariate t-distribution are given in Section 3. Section 4 is a conclusion.

A method to produce a multivariate pdf with known scale matrix

Consider the transformation

The scale matrix

For the

From linear algebra,

random variables

To create a multivariate normal pdf, start with the joint pdf

where

The requirement for zero mean random variables is not a restriction. If

Use Equation (2) to transform the variables. The Jacobian determinant of the transformation relates the products of the infinitesimals of integration such that

The magnitude of the Jacobian determinant of the transformation

where the equality

Since

normally distributed variables.

The result is that the unit normal, independent, multivariate pdf, Equation (4), becomes under the trans- formation Equation (2)

where

For the

from which

The denominator in the expression for

A similar approach can be used to create a multivariate Student’s t pdf. Assume truncated or effectively truncated t-distributions, so that moments exist [

Start with the joint pdf for n independent, zero-mean (location parameters

with

Use the transformation of Equation (2) to create a multivariate pdf

The solution

matrix

From the definition of the exponential function

and

In the limit as

In this subsection some examples for the variances and covariances of a multivariate Student’s t-distribution using the

The variance of the random variable

with the limits of the integrations equal to

Perform the integrations as listed. The integral over

where the

Repeat the procedure for the integrals for

The variance of the random variable

The expression for

Truncation or effective truncation of the pdf keeps the moments finite [

which is finite provided that

In the interest of brevity, only variances and covariances that were calculated for support of

If the

The covariance

If the

The expression for

The expressions for

Given a matrix

A general expression for the covariance (assuming support

If support is

Unlike normally distributed random variables, the correlation matrix

Given a matrix of the variances and the covariances,

A multivariate Student’s t-distribution is derived by analogy to the derivation for a multivariate normal (or Gaussian) pdf. The variances and covariances for the multivariate t-distribution are given. It is noteworthy that the shape parameters

This work was funded by the Natural Science and Engineering Research Council (NSERC) Canada.

Daniel T. Cassidy, (2016) A Multivariate Student’s t-Distribution. Open Journal of Statistics,06,443-450. doi: 10.4236/ojs.2016.63040

The Jacobian determinant is used in physics, mathematics, and statistics. Many of these uses can be traced to the Jacobian determinate as a measure of the volume of an infinitesimially small, n-dimensional parallelepiped.

The volume of an n-dimensional parallelepiped is given by the absolute value of the determinant of the com- ponents of the edge vectors that form the parallelepiped.

The area of a parallelogram with edge vectors

The volume of a parallelepiped with edge vectors

Assume that there are n functions

where

To simplify the notation, assume that

These equations can be put in matrix form

These three equations can be solved for the

The Jacobian determinant of the transformation is used in change of variables in integration:

The absolute value sign is required since the determinant could be negative (i.e., the volume could decrease).

The Jacobian determinant for the inverse transformation (to obtain

which equals