#
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.

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 |

## 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*

*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.

## Related concepts

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

- 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

- 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.