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Multivariate Student's t distribution

This lecture deals with the multivariate Student's t distribution. We first introduce the special case in which the mean is equal to zero and the scale matrix is equal to the identity matrix. We then deal with the more general case.

The standard multivariate Student's t distribution

The adjective "standard" is used for a multivariate Student's t distribution having zero mean and unit scale matrix.

Definition

Standard multivariate Student's t random vectors are characterized as follows:

Definition Let X be a Kx1 absolutely continuous random vector. Let its support be the set of K-dimensional real vectors:[eq1]Let [eq2]. We say that X has a standard multivariate Student's t distribution with n degrees of freedom if its joint probability density function is:[eq3]where[eq4]and [eq5] is the Gamma function.

Relation to the univariate Student's t distribution

When $K=1$, the definition of the standard multivariate Student's t distribution coincides with the definition of the standard univariate Student's t distribution:[eq6]

Relation to the Gamma and multivariate normal distributions

A standard multivariate Student's t random vector can be written as a multivariate normal vector whose covariance matrix is scaled by the reciprocal of a Gamma random variable, as shown by the following proposition:

Proposition (Integral representation) The joint probability density function of X can be written as:[eq7]where:

  1. [eq8] is the joint probability density function of a multivariate normal distribution with mean 0 and covariance $V= rac{1}{z}I$ (where I is the $K imes K$ identity matrix):[eq9]where [eq10]

  2. [eq11] is the probability density function of a Gamma random variable with parameters n and $h=1$:[eq12]where[eq13]

Proof

We need to prove that:[eq14]where:[eq15]and[eq16]We start from the integrand function: [eq17]where [eq18]and [eq19] is the probability density function of a random variable having a Gamma distribution with parameters $n+K$ and [eq20].Therefore:[eq21]

Marginals

The marginal distribution of the i-th component of X (denote it by $X_{i} $) is a standard Student's t distribution with n degrees of freedom. It suffices to note that the marginal probability density function of X_i can be written as:[eq22]where [eq23] is the marginal density of [eq24], i.e. the density of a normal random variable with mean 0 and variance $ rac{1}{z}$:[eq25]

Proof

This is obtained by exchanging the order of integration:[eq26]But, by the above proposition (Integral representation), the fact that[eq27]implies that X_i has a standard multivariate Student's t distribution with n degrees of freedom (hence a standard Student's t distribution with n degrees of freedom, because the two are the same thing when $K=1$).

Expected value

The expected value of a standard multivariate Student's t random vector X is well-defined only when $n>1$ and it is:[eq28]

Proof

[eq29] if [eq30] for all K components X_i. But the marginal distribution of X_i is a standard Student's t distribution with n degrees of freedom. Therefore: [eq31]provided $n>1$.

Covariance matrix

The covariance matrix of a standard multivariate Student's t random vector X is well-defined only when $n>2$ and it is:[eq32]where I is the $K imes K$ identity matrix.

Proof

Using the formula for computing the covariance matrix and the fact that [eq33] and [eq34], we obtain:[eq35]Using the two equations above and the fact that [eq36], we obtain:[eq37]But [eq38] . Therefore: [eq39]where [eq40] has been obtained as follows:[eq41]

The multivariate Student's t distribution in general

While in the previous section we restricted our attention to the multivariate Student's t distribution with zero mean and unit scale matrix, we now deal with the general case.

Definition

Multivariate Student's t random vectors are characterized as follows:

Definition Let X be a Kx1 absolutely continuous random vector. Let its support be the set of K-dimensional real vectors:[eq42]Let mu be a Kx1 vector, V a $K imes K$ symmetric and positive definite matrix and [eq2]. We say that X has a multivariate Student's t distribution with mean mu, scale matrix V and n degrees of freedom if its joint probability density function is:[eq44]where[eq45]

We indicate that X has a multivariate Student's t distribution with mean mu, scale matrix V and n degrees of freedom by:[eq46]

Relation between standard and general

If [eq47], then X is a linear function of a standard Student's t random vector:

Proposition Let [eq47]. Then:[eq49]where Z is a Kx1 vector having a standard multivariate Student's t distribution with n degrees of freedom and Sigma is a $K imes K$ invertible matrix such that [eq50].

Proof

This is proved using the formula for the joint density of a linear function of an absolutely continuous random vector ([eq51] is a linear one-to-one mapping since Sigma is invertible):[eq52]The existence of a matrix Sigma satisfying [eq53] is guaranteed by the fact that V is symmetric and positive definite.

Expected value

The expected value of a multivariate Student's t random vector X is:[eq54]

Proof

This is an immediate consequence of the fact that $X=mu +Sigma Z$ (where Z has a standard multivariate Student's t distribution) and of the linearity of the expected value:[eq55]

Covariance matrix

The covariance matrix of a multivariate Student's t random vector X is:[eq56]

Proof

This is an immediate consequence of the fact that $X=mu +Sigma Z$ (where Z has a standard multivariate Student's t distribution) and of the Addition to constant vectors and Multiplication by constant matrices properties of the covariance matrix:[eq57]