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-distribution, for the case of p dimensions, is based on the observation that if \mathbf y and u are independent and distributed as and \chi^2_\nu (i.e. multivariate normal and chi-squared distributions) respectively, the matrix is a p × p matrix, and is a constant vector then the random variable has the density and is said to be distributed as a multivariate t-distribution with parameters. Note that is not the covariance matrix since the covariance is given by (for \nu>2). The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm: This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals: where indicates a gamma distribution with density proportional to, and conditionally follows. In the special case \nu=1, the distribution is a multivariate Cauchy distribution.

Derivation

There are in fact many candidates for the multivariate generalization of Student's t-distribution. An extensive survey of the field has been given by Kotz and Nadarajah (2004). The essential issue is to define a probability density function of several variables that is the appropriate generalization of the formula for the univariate case. In one dimension (p=1), with t=x-\mu and \Sigma=1, 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 theory, where one writes down a corresponding function of p variables t_i that replaces t^2 by a quadratic function of all the t_i. It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom \nu. With, 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: Note that. Now, if \mathbf{A} is the identity matrix, the density is The difficulty with the standard representation is revealed by this formula, which does not factorize into the product of the marginal one-dimensional distributions. When \Sigma 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 (cdf) in one dimension can be extended to multiple dimensions by defining the following probability (here \mathbf{x} is a real vector): There is no simple formula for, 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. Let vector X follow a multivariate t distribution and partition into two subvectors of p_1, p_2 elements: where, the known mean vectors are and the scale matrix is. Roth and Ding find the conditional distribution p(X_1|X_2) to be a new t-distribution with modified parameters. An equivalent expression in Kotz et. al. is somewhat less concise. Thus the conditional distribution is most easily represented as a two-step procedure. Form first the intermediate distribution above then, using the parameters below, the explicit conditional distribution becomes where

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, take the simplest centralised case with spherical symmetry and no scaling,, then the multivariate t -PDF takes the form where and \nu = degrees of freedom as defined in Muirhead section 1.5. The covariance of X is The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder, define radial measure and, noting that the density is dependent only on r2, we get""which is equivalent to the variance of p-element vector X treated as a univariate heavy-tail zero-mean random sequence with uncorrelated, yet statistically dependent, elements.

Radial Distribution

follows the Fisher-Snedecor or F distribution: having mean value. F-distributions arise naturally in tests of sums of squares of sampled data after normalization by the sample standard deviation. By a change of random variable to in the equation above, retaining p-vector X, we have and probability distribution which is a regular Beta-prime distribution having mean value.

Cumulative Radial Distribution

Given the Beta-prime distribution, the radial cumulative distribution function of y is known: where I is the incomplete Beta function and applies with a spherical \Sigma assumption. In the scalar case, p = 1, the distribution is equivalent to Student-t with the equivalence, the variable t having double-sided tails for CDF purposes, i.e. the "two-tail-t-test". The radial distribution can also be derived via a straightforward coordinate transformation from Cartesian to spherical. A constant radius surface at with PDF is an iso-density surface. Given this density value, the quantum of probability on a shell of surface area A_R and thickness \delta R at R is. The enclosed p-sphere of radius R has surface area. Substitution into \delta P shows that the shell has element of probability which is equivalent to radial density function which further simplifies to where B(,) is the Beta function. Changing the radial variable to y=R^2 / \nu returns the previous Beta Prime distribution To scale the radial variables without changing the radial shape function, define scale matrix, yielding a 3-parameter Cartesian density function, ie. the probability \Delta_P in volume element is or, in terms of scalar radial variable R,

Radial Moments

The moments of all the radial variables, with the spherical distribution assumption, can be derived from the Beta Prime distribution. If then, a known result. Thus, for variable we have The moments of are while introducing the scale matrix yields Moments relating to radial variable R are found by setting and M=2m whereupon

Linear Combinations and Affine Transformation

Full Rank Transform

This closely relates to the multivariate normal method and is described in Kotz and Nadarajah, Kibria and Joarder, Roth, and Cornish. Starting from a somewhat simplified version of the central MV-t pdf:, where \Kappa is a constant and \nu is arbitrary but fixed, let be a full-rank matrix and form vector. Then, by straightforward change of variables The matrix of partial derivatives is and the Jacobian becomes. Thus The denominator reduces to In full: which is a regular MV-t distribution. In general if and has full rank p then

Marginal Distributions

This is a special case of the rank-reducing linear transform below. Kotz defines marginal distributions as follows. Partition into two subvectors of p_1, p_2 elements: with, means , scale matrix then, such that If a transformation is constructed in the form then vector, as discussed below, has the same distribution as the marginal distribution of X_1.

Rank-Reducing Linear Transform

In the linear transform case, if \Theta is a rectangular matrix, of rank m the result is dimensionality reduction. Here, Jacobian is seemingly rectangular but the value in the denominator pdf is nevertheless correct. There is a discussion of rectangular matrix product determinants in Aitken. In general if and has full rank m then In extremis, if m = 1 and \Theta becomes a row vector, then scalar Y follows a univariate double-sided Student-t distribution defined by with the same \nu degrees of freedom. Kibria et. al. use the affine transformation to find the marginal distributions which are also MV-t.

Related concepts

Literature