Wishart distribution and their properties, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

2.1 The Wishart distribution

The Wishart distribution is a family of distributions for symmetric positive definite matrices. Let X₁,...,X_n be independent N_p(0,Σ) and form a p×n data matrix X = [X₁,...,X_n]. The distribution of a p × p random matrix  is said to have the Wishart distribution.

Definition 1. The random matrix  has the Wishart distribution with n degrees of freedom and covariance matrix Σ and is denoted by M ∼ W_p(n,Σ). For n ≥ p, the probability density function of M is

with respect to Lebesque measure on the cone of symmetric positive definite matrices. Here, Γ_p(α) is the multivariate gamma function

The precise form of the density is rarely used. Two exceptions are that i) in Bayesian computation, the Wishart distribution is often used as a conjugate prior for the inverse of normal covariance matrix and that ii) when symmetric positive definite matrices are the random elements of interest in diffusion tensor study.

The Wishart distribution is a multivariate extension of χ2 distribution. In particular, if  is called the standard Wishart distribution. 

The law of large numbers and the Cramer-Wold device leads to M_n/n → Σ in probabilityas n →∞.

Corollary 2. If M ∼ W_p(n,Σ) and a ∈R^p is such that , then 

Theorem 3. If M ∼ W_p(n,Σ) and a ∈R^p and n > p−1, then 

 The previous theorem holds for any deterministic a ∈ R^p, thus holds for any randoma provided that the distribution of a is independent of M. This is important in the next subsection.

The following lemma is useful in a proof of Theorem 3   

2.2 Hotelling’s T² statistic

Definition 2. Suppose X and S are independent and such that 

Then

is known as Hotelling’s T² statistic.

Hotelling’s T² statistic plays a similar role in multivariate analysis to that of the student’s t-statistic in univariate statistical analysis. That is, the application of Hotelling’s T² statistic is of great practical importance in testing hypotheses about the mean of a multivariate normal distribution when the covariance matrix is unknown. 

Theorem 5. If m > p−1,

A special case is when p = 1, where Theorem 5 indicates that  Where isthe connection of the Hotelling’s T² statistic to the student’s t-distribution? Note that we are indeed abusing the definition of ‘statistic’ here.

2.3 Samples from a multivariate normal distribution

Suppose X₁,...,X_n are i.i.d. N_p(µ,Σ). Denote the sample mean and sample variance by

Theorem 6. and S are independent, with 

Corollary 7. The Hotelling’s T² statistic for MVN sample is defined as 

and we hav

(Incomplete) proof of Theorem 6. First note the following decomposition: 

and recall the definition of the Wishart distribution.

It is easy to check the result on . The following argument is for the indepen-dence and the distribution of S.

Consider a new set of random vectors Y_i (i = 1,...,n) from a linear combination of X_is by an orthogonal matrix D satisfying

Let

where   is the p×n matrix of de-meaned random vectors. We claim thefollowing:

1. Y_j are normally distributed.

The facts 1 and 2 show the independence, while facts 3 and 4 give the distribution of S.

Next lecture is on the inference about the multivariate normal distribution.