Wishart distribution

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The joint distribution of the elements from the sample covariance matrix of observations from a multivariate normal distribution. Let the results of observations have a -dimensional normal distribution with vector mean and covariance matrix . Then the joint density of the elements of the matrix is given by the formula

( denotes the trace of a matrix ), if the matrix is positive definite, and in other cases. The Wishart distribution with degrees of freedom and with matrix is defined as the -dimensional distribution with density . The sample covariance matrix , which is an estimator for the matrix , has a Wishart distribution.

The Wishart distribution is a basic distribution in multivariate statistical analysis; it is the -dimensional generalization (in the sense above) of the -dimensional "chi-squared" distribution.

If the independent random vectors and have Wishart distributions and , respectively, then the vector has the Wishart distribution .

The Wishart distribution was first used by J. Wishart [1].


[1] J. Wishart, Biometrika A , 20 (1928) pp. 32–52
[2] T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1958)



[a1] A.M. Khirsagar, "Multivariate analysis" , M. Dekker (1972)
[a2] R.J. Muirhead, "Aspects of multivariate statistical theory" , Wiley (1982)
How to Cite This Entry:
Wishart distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article