Wishart distribution

In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables ("random matrices"), defined as follows. Suppose

i.e. X₁ is a p×1 column-vector-valued random variable (a "random vector") that is normally distributed, whose expected value is the p×1 column vector whose entries are all zero, and whose variance is the p×p nonnegative definite matrix V. We have

and

where the transpose of any matrix A is denoted A′.

Further suppose X₁, ..., X_n are independent and identically distributed. Then the Wishart distribution is the probability distribution of the p×p random matrix

One indicates that S has that probability distribution by writing

The positive integer n is the number of degrees of freedom.

If p = 1 and V = 1 then this distribution is a chi-square distribution.

The Wishart distribution arises frequently in likelihood-ratio tests in multivariate statistical analysis.