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.
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This section introduces the simpler, but less general, "standard" case.
Standard 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 . We say that has a standard multivariate Student's t distribution with degrees of freedom if its joint probability density function iswhereand is the Gamma function.
When , the definition of the standard multivariate Student's t distribution coincides with the definition of the standard univariate Student's t distribution.
This is proved as follows:The latter is the probability density function of a standard univariate Student's t distribution.
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 can be written aswhere:
is the joint probability density function of a multivariate normal distribution with mean and covariance (where is the identity matrix):where
is the probability density function of a Gamma random variable with parameters and :where
We need to prove thatwhereandWe start from the integrand function: where and is the probability density function of a random variable having a Gamma distribution with parameters and . Therefore,
The marginal distribution of any one of the entries of is a univariate standard Student's t distribution with degrees of freedom.
Denote the -th component of by . The marginal probability density function of is derived by integrating the joint probability density function with respect to the other entries of :whereis the marginal probability density function of the entry of a multivariate normal random vector with mean and covariance . This is equal to the density of a normal random variable with mean and variance :Therefore, we have thatBut, by the above proposition (Integral representation), this implies that has a standard multivariate Student's t distribution with degrees of freedom. Hence, it has a standard univariate Student's t distribution with degrees of freedom, because the two are the same thing when .
The expected value of a standard multivariate Student's t random vector is well-defined only when and it is
Note that if for all components . But the marginal distribution of is a standard Student's t distribution with degrees of freedom. Therefore, provided .
The covariance matrix of a standard multivariate Student's t random vector is well-defined only when and it iswhere is the identity matrix.
We have proved above that . This impliesWe have also proved that has a multivariate normal distribution with meanand covariance matrixAs a consequence,andwhere has been obtained as follows:
This section deals with the general case.
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
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 .
Because is invertible, we have thatis a linear one-to-one mapping. Therefore, we can use the formula for the joint density of a linear function of a continuous random vector:The existence of a matrix satisfying is guaranteed by the fact that is symmetric and positive definite.
The expected value of a multivariate Student's t random vector is
This is an immediate consequence of the fact that (where has a standard multivariate Student's t distribution) and of the linearity of the expected value:
The covariance matrix of a multivariate Student's t random vector is
This is an immediate consequence of the fact that (where 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:
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