|Parameters|| location (real vector)
scale matrix (positive-definite real matrix)
is the degrees of freedom
|CDF||No analytic expression|
|Mean||if ; else undefined|
|Variance||if ; else undefined|
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
and is said to be distributed as a multivariate t-distribution with parameters .
In the special case , the distribution is a multivariate Cauchy distribution.
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 (), with and , we have the probability density function
and one approach is to write down a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of variables that replaces by a quadratic function of all the . It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom . 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 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 is diagonal the standard representation can be shown to have zero correlation but the marginal distributions do not agree with statistical independence. There are differing views on this issue, which is under discussion in the research literature as of early 2007.
Many such distributions may be constructed by considering the quotients of normal random variables with the square root of a sample from a chi-squared distribution. These are surveyed in the references and links below.
Copulas based on the multivariate t
In univariate statistics, the Student's t-test makes use of Student's t-distribution. 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.
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (May 2012)|
- Copula Methods vs Canonical Multivariate Distributions: the multivariate Student T distribution with general degrees of freedom
- Multivariate Student's t distribution