Multivariate t-distribution on arvau

Wikipedia / Multivariate t-distribution

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.

Definition

One common method of construction of a multivariate t (i.e. multivariate normal and chi-squared distributions is a pp has the density

and is said to be distributed as a multivariate t).

The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm:

, independently.
.

This formulation gives rise to the hierarchical representation of a multivariate t.

, the distribution is a multivariate Cauchy distribution.

Derivation

There are in fact many candidates for the multivariate generalization of Student's t-distribution, we have the probability density function and one approach is to use a corresponding function of several variables. This is the basic idea of elliptical distribution. It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom, one has a simple choice of multivariate density function

which is the standard but not the only choice.

An important special case is the standard bivariate t-distribution, p = 2:

is the identity matrix, the density is

is diagonal the standard representation can be shown to have zero correlation but the marginal distributions are not statistically independent.

A notable spontaneous occurrence of the elliptical multivariate distribution is its formal mathematical appearance when least squares methods are applied to multivariate normal data such as the classical Markowitz minimum variance econometric solution for asset portfolios.

Cumulative distribution function

The definition of the cumulative distribution function is a real vector):

, but it can be approximated numerically via Monte Carlo integration.

Conditional Distribution

This was developed by Muirhead and Cornish, but later derived using the simpler chi-squared ratio representation above, by Roth and Ding. follow a multivariate t

to be a new t-distribution with modified parameters.

An equivalent expression in Kotz et al. is somewhat less concise.

above then, using the parameters below, the explicit conditional distribution becomes

is the Schur complement is the squared Mahalanobis distance.

Copulas based on the multivariate t

The use of such distributions is enjoying renewed interest due to applications in mathematical finance, especially through the use of the Student's t copula.

Elliptical representation

Constructed as an elliptical distribution,, then the multivariate <big>t</big>-PDF takes the form

is the degrees of freedom as defined in Muirhead is

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder, and, noting that the density is dependent only on r2, we get treated as a univariate heavy-tail zero-mean random sequence with uncorrelated, yet statistically dependent, elements.

Radial Distribution

follows the Fisher-Snedecor distribution:

-distributions arise naturally in tests of sums of squares of sampled data after normalization by the sample standard deviation.

which is a regular Beta-prime distribution.

Cumulative Radial Distribution

is the incomplete Beta function assumption.

, the distribution is equivalent to Student-t, the variable t having double-sided tails for CDF purposes, i.e. the "two-tail-t-test".

is the Beta function.

Radial Moments

Linear Combinations and Affine Transformation

Full Rank Transform

. Then, by straightforward change of variables

which is a regular MV-t distribution.

Marginal Distributions

Rank-Reducing Linear Transform

in the denominator pdf is nevertheless correct. There is a discussion of rectangular matrix product determinants in Aitken. then

In extremis, if m becomes a row vector, then scalar Y degrees of freedom. Kibria et al. use the affine transformation to find the marginal distributions which are also MV-t.

whose elements remain 'entangled' and are not statistically independent.
A vector of independent student-t samples is not consistent with the multivariate t distribution.
Adding two sample multivariate t will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.
Taleb compares many examples of fat-tail elliptical vs non-elliptical multivariate distributions

Related concepts

In univariate statistics, the Student's t-test makes use of Student's t-distribution
The elliptical multivariate-t distribution arises spontaneously in linearly constrained least squares solutions involving multivariate normal source data, for example the Markowitz global minimum variance solution in financial portfolio analysis. which addresses an ensemble of normal random vectors or a random matrix. It does not arise in ordinary least squares (OLS) or multiple regression with fixed dependent and independent variables which problem tends to produce well-behaved normal error probabilities.
Hotelling's T-squared distribution is a distribution that arises in multivariate statistics.
The matrix t-distribution is a distribution for random variables arranged in a matrix structure.