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 .
|||J. Wishart, Biometrika A , 20 (1928) pp. 32–52|
|||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)|
Wishart distribution. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Wishart_distribution&oldid=28557