A Multivariate Student ’ s t-Distribution

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 i ν 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.


Introduction
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 ν for each marginal probability density function (pdf) of the multivariate pdf to be different.
The form that is typically used is [1] ( ) ( ) 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 ν .The shape of this multivariate t-distribution arises from the observation that the pdf for [ ] [ ] σ σ is distributed as chi-squared.
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.

Background Information
(2) is the expectation of i j y y and , , i j j i , where the superscript T indicates a transpose of the matrix.

Multivariate Normal Probability Density Function
To create a multivariate normal pdf, start with the joint pdf [ ] ( ) where [ ] gives the probability that The requirement for zero mean random variables is not a restriction.If { } x x is a zero mean random variable with the same shape and scale parameters as i x .Use Equation (2) to transform the variables.The Jacobian determinant of the transformation relates the products of the infinitesimals of integration such that ( ) ( ) x y y y y y y The magnitude of the Jacobian determinant of the where the equality The result is that the unit normal, independent, multivariate pdf, Equation ( 4), becomes under the transformation Equation ( 2) where [ ] The denominator in the expression for

Multivariate Student's t Probability Density Function
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 [3] ( ) gives the probability that a random draw of the column matrix . The pdf ( ) x and the shape parameter i ν , and thus is independent of any other ( ) Use the transformation of Equation ( 2) to create a multivariate pdf The solution ) and ; f y ν t , Equation (11), becomes a multivariate normal distribution.

Σ for the n 4 = Example
In this subsection some examples for the variances and covariances of a multivariate Student's t-distribution using the 4 n = example of Equation ( 2) are given.The variance of the random variable 3 y is which is finite provided that b < .
In the interest of brevity, only variances and covariances that were calculated for support of [ ] , −∞ ∞ will be discussed.The requirement that 2 i ν > will be understood to be waived if the pdf is truncated or effectively truncated.It is also to be understood that the variances and covariances as calculated for support of [ ] , −∞ ∞ provide upper limits for variances and covariances calculated for truncation or effective truncation of the pdf.
If the i ν are not equal, then for the The covariance { } = for all i is given by { } ( ) If the i ν are not equal, then the covariance { } The expression for { } E y y , which is valid for the i ν not equal, is { } The expressions for { }   )

General Expressions for
, then the general expressions need to be multiplied by functions that depend on b and ν .Truncation or effective truncation keeps the moments finite and defined for all 1 ν ≥ [3]- [5].The general expressions for the covariance, Equation (24), yields, when i j = , the general expression for the variance, Equation (23).The general expression for the variance, Equation (23), is given to emphasize the 2 , j j m

Inversion Exists
Assume that there are n functions , , , i i n x f q q q =  .The necessary and sufficient condition that the functions can be inverted to find ( ) , , , , , , 0 , , , To simplify the notation, assume that 3 n = so that ( ) These equations can be put in matrix form These three equations can be solved for the d i q if the determinant of the 3 3 × matrix is non-zero.This is a standard result from linear algebra.The determinant of the 3 3 × matrix is called the Jacobian determinant of the transformation.

Change of Variables
The Jacobian determinant of the transformation is used in change of variables in integration: x x x V x x x q q q q q q ∂ = = ∂ ∫∫∫ ∫∫∫ ∫∫∫ (30) 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 [ ] x as functions of [ ] y ) given by Equation (8) is is given by Equation (1) when [ ] y is distributed as a multivariate normal distribution with covariance matrix [ ] Σ and 2

2. 1 .x are 4 1 ×M is 4 4 ×
Cholesky Decomposition A method to produce a multivariate pdf with known scale matrix [ ] s Σ is presented in this section.For normally distributed variables, the covariance matrix [ ] [ ] s Σ = Σ since the scale factor for a normal distribution is the standard deviation of the distribution.An example with 4 n = is used to provide concrete examples.Consider the transformation [ ] [ ][ ] column matrices, [ ] square matrix, and the elements of [ ] x are independent random variables.The off-diagonal elements of [ ] M introduce correlations between the elements of [ ] y .

From
[4].For simplicity, assume that support is [ ] , large number, β is the scale factor for the distribution, and µ is the location parameter for the distribution.If b is a large number, then a significant portion of the tails of the distribution are included.If b = ∞ then all of the tails are included.Start with the joint pdf for n independent, zero-mean (location parameters [ ] 0 µ = ) Student's t pdfs with shape parameters [ ] ν , and scale parameters [ ] 1 Equation (2) was used.The elements of the inverse matrix [ ] (8) for the 4 n = example.Note that the shape parameters i ν of the constituent distributions need not be the same in the multivariate t-distribution given by [ ] [ ] (

1 ; 4 µ
− are the elements of the inverse of matrix [ ] M and are as given by [ ] 1 M − , Equation (8), and is a constant as far as the integral over 4 y is concerned.Repeat the procedure for the integrals for integrals are not equal to unity owing to the presence of the 2 The variance of the random variable i y for the multivariate Student's t-distribution with support [ ] where i β is a scale factor and b < ∞ [3]-[5].Note that the scale factors for the multivariate t-distribution are , truncation of the pdf keeps the moments finite [3]-[5].For example, the second central moment for a 1 ν = Student's t-distribution with scale factor β and support [ ] , b b

3 3 E y y , { } 2 3 E 3 E
y y , and { } 1 y y show a simple pattern for the relationship between the covariance matrix Σ , the scale matrix [ ] s Σ Equation (3), and the matrix [ ] M Equation (2).
all i, and i n ≤ ) for the multivariate Student's t-distribution