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Notation | |||
---|---|---|---|
Parameters | location (real matrix) Contentsdegrees of freedom (real) | ||
Support | |||
CDF | No analytic expression | ||
Mean | if , else undefined | ||
Mode | |||
Variance | if , else undefined | ||
CF | see below |
In statistics, the matrix t-distribution (or matrix variate t-distribution) is the generalization of the multivariate t-distribution from vectors to matrices. [1] [2]
The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse Wishart distribution placed over either of its covariance matrices, [1] and the multivariate t-distribution can be generated in a similar way. [2]
In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix t-distribution is the posterior predictive distribution. [3]
For a matrix t-distribution, the probability density function at the point of an space is
where the constant of integration K is given by
Here is the multivariate gamma function.
The characteristic function and various other properties can be derived from the generalized matrix t-distribution (see below).
Notation | |||
---|---|---|---|
Parameters | location (real matrix) | ||
Support | |||
| |||
CDF | No analytic expression | ||
Mean | if , else undefined | ||
Variance | if , else undefined | ||
CF | see below |
The generalized matrix t-distribution is a generalization of the matrix t-distribution with two parameters and in place of . [3]
This reduces to the standard matrix t-distribution with
The generalized matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse multivariate gamma distribution placed over either of its covariance matrices.
The property above comes from Sylvester's determinant theorem:
If and and are nonsingular matrices then [2] [3]
The characteristic function is [3]
where
and where is the type-two Bessel function of Herz[ clarification needed ] of a matrix argument.
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