# Wishart distribution

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].

#### References

[1] | J. Wishart, Biometrika A , 20 (1928) pp. 32–52 |

[2] | T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1958) |

#### Comments

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Wishart_distribution&oldid=28557