Wishart distribution

The joint distribution of the elements from the sample covariance matrix of observations from a multivariate normal distribution. Let the results of observations have a $p$-dimensional normal distribution $N(\mu ,\mathrm{\Sigma })$ with vector mean $\mu$ and covariance matrix $\mathrm{\Sigma }$. Then the joint density of the elements of the matrix $A=\sum _{i=1}^n(X_i-\bar{X})(X_i-\bar{X}{)}^{\prime }$ is given by the formula

$$w(n,\mathrm{\Sigma })=\frac{|A{|}^{(n-p)/2}{e}^{-\mathrm{tr}(A{\mathrm{\Sigma }}^{-1})/2}}{2^{(n-1)p/2}{\pi }^{p(p-1)/4}|\mathrm{\Sigma }{|}^{(n-1)/2}\prod _{i=1}^p\mathrm{\Gamma }(n-\frac{i}{2})}$$

( $\mathrm{tr}M$ denotes the trace of a matrix $M$), if the matrix $\mathrm{\Sigma }$ is positive definite, and $w(n,\mathrm{\Sigma })=0$ in other cases. The Wishart distribution with $n$ degrees of freedom and with matrix $\mathrm{\Sigma }$ is defined as the $p(n+1)/2$- dimensional distribution $W(n,\mathrm{\Sigma })$ with density $w(n,\mathrm{\Sigma })$. The sample covariance matrix $S=A/(n-1)$, which is an estimator for the matrix $\mathrm{\Sigma }$, has a Wishart distribution.

The Wishart distribution is a basic distribution in multivariate statistical analysis; it is the $p$-dimensional generalization (in the sense above) of the $1$-dimensional "chi-squared" distribution.

If the independent random vectors $X$ and $Y$ have Wishart distributions $W(n_1,\mathrm{\Sigma })$ and $W(n_2,\mathrm{\Sigma })$, respectively, then the vector $X+Y$ has the Wishart distribution $W(n_1+n_2,\mathrm{\Sigma })$.

The Wishart distribution was first used by J. Wishart [1].

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