Multivariate Student's t distribution | Properties and proofs (original) (raw)

The multivariate (MV) Student's t distribution is a multivariate continuous distribution that generalizes the one-dimensional Student's t distribution.

Table of contents

  1. How the distribution is derived
  2. The standard multivariate Student's t distribution
    1. Definition
    2. Relation to the univariate Student's t distribution
    3. Relation to the Gamma and multivariate normal distributions
    4. Marginals
    5. Expected value
    6. Covariance matrix
  3. The multivariate Student's t distribution in general
    1. Definition
    2. Relation between standard and general
    3. Expected value
    4. Covariance matrix

How the distribution is derived

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 multivariate Student vectors.

Multivariate normal divided by square root of Gamma equals standard Student.

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.

Relation to the univariate Student's t distribution

When, 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.

Proof

Marginals

The marginal distribution of any one of the entries of is a univariate standard Student's t distribution with degrees of freedom.

Proof

Expected value

The expected value of a standard multivariate Student's t random vector is well-defined only when and it is

Proof

Covariance matrix

The covariance matrix of a standard multivariate Student's t random vector is well-defined only when and it iswhere is the identity matrix.

Proof

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 be a continuous random vector. Let its support be the set of-dimensional real vectors:Let be a vector, a symmetric and positive definite matrix and. We say that has a multivariate Student's t distribution with mean,scale matrix and degrees of freedom if its joint probability density function iswhere

We indicate that has a multivariate Student's t distribution with mean, scale matrix and degrees of freedom by

Relation between standard and general

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

Proposition Let. Then,where is a vector having a standard multivariate Student's t distribution with degrees of freedom and is a invertible matrix such that.

Proof

Expected value

The expected value of a multivariate Student's t random vector is

Proof

Covariance matrix

The covariance matrix of a multivariate Student's t random vector is

Proof

How to cite

Please cite as:

Taboga, Marco (2021). "Multivariate Student's t distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/multivariate-student-t-distribution.