It is named in honor of John Wishart, who first formulated the distribution in 1928.

[1] Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).

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

In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector.

One indicates that S has that probability distribution by writing The positive integer n is the number of degrees of freedom.

It occurs frequently in likelihood-ratio tests in multivariate statistical analysis.

It also arises in the spectral theory of random matrices[citation needed] and in multidimensional Bayesian analysis.

[6] The Wishart distribution can be characterized by its probability density function as follows: Let X be a p × p symmetric matrix of random variables that is positive semi-definite.

Let V be a (fixed) symmetric positive definite matrix of size p × p. Then, if n ≥ p, X has a Wishart distribution with n degrees of freedom if it has the probability density function where

Also, the density formula above applies only to positive definite matrices

If n ≤ p − 1, then the Wishart no longer has a density—instead it represents a singular distribution that takes values in a lower-dimension subspace of the space of p × p matrices.

[10] In Bayesian statistics, in the context of the multivariate normal distribution, the Wishart distribution is the conjugate prior to the precision matrix Ω = Σ−1, where Σ is the covariance matrix.

[11]: 135 [12] The least informative, proper Wishart prior is obtained by setting n = p.[citation needed] A common choice for V leverages the fact that the mean of X ~Wp(V, n) is nV.

Then V is chosen so that nV equals an initial guess for X.

For instance, when estimating a precision matrix Σ−1 ~ Wp(V, n) a reasonable choice for V would be n−1Σ0−1, where Σ0 is some prior estimate for the covariance matrix Σ.

The following variance computation could be of help in Bayesian statistics: where

This comes up when computing the Fisher information of the Wishart random variable.

is The characteristic function of the Wishart distribution is where E[⋅] denotes expectation.

[9] Properly interpreting this formula requires a little care, because noninteger complex powers are multivalued; when n is noninteger, the correct branch must be determined via analytic continuation.

— and C is a q × p matrix of rank q, then [15] If z is a nonzero p × 1 constant vector, then:[15] In this case,

Then corollary 1 above shows that gives the marginal distribution of each of the elements on the matrix's diagonal.

Seber prefers to reserve the term multivariate for the case when all univariate marginals belong to the same family.

The Bartlett decomposition of a matrix X from a p-variate Wishart distribution with scale matrix V and n degrees of freedom is the factorization: where L is the Cholesky factor of V, and: where

[18] This provides a useful method for obtaining random samples from a Wishart distribution.

[19] Let V be a 2 × 2 variance matrix characterized by correlation coefficient −1 < ρ < 1 and L its lower Cholesky factor: Multiplying through the Bartlett decomposition above, we find that a random sample from the 2 × 2 Wishart distribution is The diagonal elements, most evidently in the first element, follow the χ2 distribution with n degrees of freedom (scaled by σ2) as expected.

The off-diagonal element is less familiar but can be identified as a normal variance-mean mixture where the mixing density is a χ2 distribution.

The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution where Kν(z) is the modified Bessel function of the second kind.

, the off-diagonal elements It is also possible to write down the moment-generating function even in the noncentral case (essentially the nth power of Craig (1936)[22] equation 10) although the probability density becomes an infinite sum of Bessel functions.

It can be shown [23] that the Wishart distribution can be defined if and only if the shape parameter n belongs to the set This set is named after Gindikin, who introduced it[24] in the 1970s in the context of gamma distributions on homogeneous cones.

However, for the new parameters in the discrete spectrum of the Gindikin ensemble, namely, the corresponding Wishart distribution has no Lebesgue density.