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.

Multivariate t
Notation
Parameters location (real vector)
shape 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

Definition

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 matrix is a p × p matrix, and , then 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 ).

In the special case , 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 (), 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

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

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

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.

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

References

  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

  • Kotz, Samuel; Nadarajah, Saralees (2004). Multivariate t Distributions and Their Applications. Cambridge University Press. ISBN 978-0521826549.
  • Cherubini, Umberto; Luciano, Elisa; Vecchiato, Walter (2004). Copula methods in finance. John Wiley & Sons. ISBN 978-0470863442.
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