In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first formulated the distribution in 1928. 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 involves Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE). It is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). 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.
Definition Suppose G is a p × n matrix, each column of which is independently drawn from a p-variate normal distribution with zero mean:
G = ( g 1 , … , g n ) ∼ N p ( 0 , V ) . {\displaystyle G=(g_{1},\dots ,g_{n})\sim {\mathcal {N}}_{p}(0,V).}
It means g i = ( g i , 1 , … , g i , p ) T ∼ i i d N p ( 0 , V ) ∀ i ∈ { 1 , … , n } {\displaystyle g_{i}=(g_{i,1},\dots ,g_{i,p})^{T}\ {\overset {iid}{\sim }}\ {\mathcal {N}}_{p}(0,V)\ \forall i\in \{1,\dots ,n\}}
Then the Wishart distribution is the probability distribution of the p × p random matrix
S = G G T = ∑ i = 1 n g i g i T {\displaystyle S=GG^{T}=\sum _{i=1}^{n}g_{i}g_{i}^{T}}
known as the scatter matrix. One indicates that S has that probability distribution by writing
S ∼ W p ( V , n ) . {\displaystyle S\sim W_{p}(V,n).}
The positive integer n is the number of degrees of freedom. Sometimes this is written W(V, p, n). For n ≥ p the matrix S is invertible with probability 1 if V is invertible. If p = V = 1 then this distribution is a chi-squared distribution with n degrees of freedom.
Occurrence The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a multivariate normal distribution. It occurs frequently in likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of random matrices and in multidimensional Bayesian analysis. It is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels.
Probability density function
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
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