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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"). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics.

Contents 1 Definition 2 Occurrence 3 Probability density function 4 Characteristic Function 5 Theorem 5.1 Corollary 1 5.2 Corollary 2 6 Estimator of the multivariate normal distribution 7 See also

Suppose X is an matrix each row of which is drawn from p-variate normal distribution with zero mean:

Further suppose that the rows X₍₁₎, ..., X_(n) are independent. Then the Wishart distribution is the probability distribution of the p×p random matrix

S = X^TX.

One indicates that S has that probability distribution by writing

The positive integer n is the number of degrees of freedom. Sometimes this is written W(V,p,n).

If p = 1 and V = 1 then this distribution is a chi-square distribution with n degrees of freedom.

The Wishart distribution arises frequently in likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of random matrices.

The Wishart distribution can be characterized by its probability density function, as follows.

Let be a symmetric matrix of random variables that is positive definite. Let be a (fixed) positive definite matrix of size .

Then, if , has a Wishart distribution with n degrees of freedom if it has a probability density function given by

where is the multivariate gamma function defined as

In fact the above definition can be extended to any real n > p − 1.

The characteristic function of the Wishart distribution is

In other words,

where denotes expectation.

If has a Wishart distribution with m degrees of freedom and variance matrix ---write ---and is a matrix of rank q, then

If is a nonzero constant vector, then .

In this case, is the chi-square distribution and (note that is a constant; it is positive because is positive definite).

Consider the case where (that is, the j-th element is one and all others zero). Then corollary 1 above shows that

gives the marginal distribution of each of the elements on the matrix's diagonal.

Noted statistician George Seber points out that the Wishart distribution is not called the "multivariate chi-square distribution" because the marginal distribution of the off-diagonal elements is not chi-square. Seber prefers to reserve the term multivariate for the case when all univariate marginals belong to the same family.

The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution. The derivation of the MLE is perhaps surprisingly subtle and elegant. It involves the spectral theorem and the reason why it can be better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices.

• Estimation of covariance matrices
• Hotelling's T-square distribution

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