Multivariate t-distribution
Multivariate t
Notation |
 |
Parameters |
location (real vector)
covariance matrix (positive-definite real matrix)
is the degrees of freedom
|
Support |
 |
PDF |
![{\displaystyle {\frac {\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{\rm {T}}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8f5dfeaf2441f91617b39b38762315c104ae6f) |
CDF |
No analytic expression, but see text for approximations |
Mean |
if ; else undefined |
Median |
 |
Mode |
 |
Variance |
if ; else undefined |
Skewness |
0 |
In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure.
One common method of construction of a multivariate t-distribution, for the case of
dimensions, is based on the observation that if
and
are independent and distributed as
and
(i.e. multivariate normal and chi-squared distributions) respectively, the covariance
is a p × p matrix, and
, then
has the density
...
Wikipedia