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, ..., Xn 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.
Referenced By
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