Wishart

Wishart is a surname of Scottish origin. In statistics, the term Wishart typically refers to the Wishart distribution, a multivariate probability distribution named after John Wishart. This distribution describes the distribution of a p-by-p positive definite random matrix that can be expressed as the sum of n independent outer products of p-dimensional Gaussian vectors with zero mean and covariance Sigma: S = sum_{i=1}^n g_i g_i^T, where g_i ~ N_p(0, Sigma). It generalizes the chi-square distribution to multiple dimensions and is denoted as W_p(n, Sigma).

The distribution is defined for integer n at least p, with Sigma a positive definite matrix. Its mean is E[S] = n Sigma. It has a closed-form density involving determinants and traces, and the multivariate gamma function. Transformations are straightforward: for any fixed p-by-p matrix A, A S A^T ~ W_p(n, A Sigma A^T).

Applications of the Wishart distribution are central in multivariate statistics. It arises as the sampling distribution of the sample covariance matrix of multivariate normal data, and thus underpins covariance estimation and hypothesis testing in multiple dimensions. In Bayesian statistics, the inverse-Wishart distribution serves as a conjugate prior for the covariance matrix. Extensions include the Wishart process, a matrix-valued stochastic process with Wishart-distributed increments, used in multivariate stochastic modeling. The distribution also appears in finance, signal processing, and machine learning for modeling and inference on covariance structures.