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Multivariate Student's t distribution

The multivariate (MV) Student's t distribution is a multivariate generalization of the one-dimensional Student's t distribution.

Recall that a random variable has a standard univariate Student's t distribution if it can be represented as a ratio between a standard normal random variable and the square root of a Gamma random variable.

Analogously, a random vector has a standard MV Student's t distribution if it can be represented as a ratio between a standard MV normal random vector and the square root of a Gamma random variable. This "standard" case is introduced in the next section, while the subsequent section deals with the more general case, that is, the case of random vectors that are obtained from linear transformations of standard MV Student's t random vectors.

The standard multivariate Student's t distribution

This section introduces the simpler, but less general, "standard" case.

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.

Proof

This is proved as follows:[eq6]The latter is the probability density function of a standard univariate Student's t distribution.

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 $frac{1}{z}I$ (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 any one of the entries of X is a univariate standard Student's t distribution with n degrees of freedom.

Proof

Denote the i-th component of X by X_i. The marginal probability density function of X_i is derived by integrating the joint probability density function with respect to the other entries of X:[eq22]where[eq23]is the marginal probability density function of the entry of a multivariate normal random vector with mean 0 and covariance $frac{1}{z}I$ . This is equal to the density of a normal random variable with mean 0 and variance $frac{1}{z}$:[eq24]Therefore, we have that[eq25]But, by the above proposition (Integral representation), this implies that X_i has a standard multivariate Student's t distribution with n degrees of freedom. Hence, it has a standard univariate 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[eq26]

Proof

Note that [eq27] if [eq28] 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, [eq29]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[eq30]where I is the $K	imes K$ identity matrix.

Proof

We have proved above that [eq27]. This implies[eq32]We have also proved that [eq33] has a multivariate normal distribution with mean[eq34]and covariance matrix[eq35]As a consequence,[eq36]and[eq37]where [eq38] has been obtained as follows:[eq39]

The multivariate Student's t distribution in general

This section deals 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:[eq40]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[eq42]where[eq43]

We indicate that X has a multivariate Student's t distribution with mean mu, scale matrix V and n degrees of freedom by[eq44]

Relation between standard and general

If [eq45], then X is a linear function of a standard Student's t random vector.

Proposition Let [eq45]. Then,[eq47]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 [eq48].

Proof

Because Sigma is invertible, we have that[eq49]is a linear one-to-one mapping. Therefore, we can use the formula for the joint density of a linear function of an absolutely continuous random vector:[eq50]The existence of a matrix Sigma satisfying [eq51] 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[eq52]

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:[eq53]

Covariance matrix

The covariance matrix 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 Addition to constant vectors and Multiplication by constant matrices properties of the covariance matrix:[eq55]

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