Multivariate t-distribution

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Multivariate t
Notation
Parameters location (real vector)
scale matrix (positive-definite real matrix)
is the degrees of freedom
Support
PDF
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.

Definition[edit]

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.

Derivation[edit]

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.

Cumulative distribution function[edit]

The definition of the cumulative distribution function (cdf) in one dimension can be extended to multiple dimensions by defining the following probability (here is a real vector):

There is no simple formula for , but it can be approximated numerically via Monte Carlo integration.[1][2]

Further theory[edit]

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[edit]

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.[citation needed]

Related concepts[edit]

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.

See also[edit]

References[edit]

  1. ^ Botev, Z. I.; L'Ecuyer, P. (6 December 2015). "Efficient probability estimation and simulation of the truncated multivariate student-t distribution". 2015 Winter Simulation Conference (WSC). Huntington Beach, CA, USA: IEEE. pp. 380–391. doi:10.1109/WSC.2015.7408180. 
  2. ^ Genz, Alan (2009). Computation of Multivariate Normal and t Probabilities. Springer. ISBN 978-3-642-01689-9. 

Literature[edit]

  • Kotz, Samuel; Nadarajah, Saralees (2004). Multivariate t Distributions and Their Applications. Cambridge University Press. ISBN 0521826543. 
  • Cherubini, Umberto; Luciano, Elisa; Vecchiato, Walter (2004). Copula methods in finance. John Wiley & Sons. ISBN 0470863447. 

External links[edit]