# 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 $N( \mu , \Sigma )$ with vector mean $\mu$ and covariance matrix $\Sigma$. Then the joint density of the elements of the matrix $A= \sum _ {i= 1} ^ {n} ( X _ {i} - \overline{X}\; ) ( X _ {i} - \overline{X}\; ) ^ \prime$ is given by the formula

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

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

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

#### References

 [1] J. Wishart, Biometrika A , 20 (1928) pp. 32–52 [2] T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1958)