Wishart distribution
In
statistics, the
Wishart distribution, named in honor of John Wishart, is any of a family of
probability distributions for nonnegativedefinite
matrixvalued
random variables ("random matrices"), defined as follows. Suppose
i.e.
X_{1} is a
p×1 columnvectorvalued 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 X_{1}, ..., 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 chisquare distribution.
The Wishart distribution arises frequently in likelihoodratio tests in multivariate statistical analysis.