Multivariate t-distribution
Notation | |
---|---|
Parameters | location (real vector) shape matrix (positive-definite real matrix) is the degrees of freedom |
Support | |
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.
Contents
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 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[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.
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See also[edit]
- Multivariate normal distribution, which is a special case of the multivariate Student's t-distribution when .
- Chi distribution, the pdf of the scaling factor in the construction the Student's t-distribution and also the 2-norm (or Euclidean norm) of a multivariate normally distributed vector (centered at zero).
- Mahalanobis distance
References[edit]
- ^ 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.
- ^ 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.