The Wishart distribution. The Wishart distribution is a useful distribution of random matrices of…

The Wishart
distribution. The Wishart distribution is a useful distribution of random
matrices of sums of squares and cross products. It can be thought of as a
multivariate generalization of the chi-squared distribution. There are various
ways of defining the Wishart distribution, each resulting in the same
distribution. A natural way to define it is in terms of transformations of a
multivariate normal distribution. Let

In the
general case, we
»

 

The Wishart
distribution. The Wishart distribution is a useful distribution of random
matrices of sums of squares and cross products. It can be thought of as a
multivariate generalization of the chi-squared distribution. There are various
ways of defining the Wishart distribution, each resulting in the same
distribution. A natural way to define it is in terms of transformations of a
multivariate normal distribution. Let

In the
general case, we denote the distribution as

The
importance of the Wishart distribution is that the Wishart matrix

can be
used to make inferences on the variance-covariance matrix _ of

an
underlying multivariate normal distribution.

a) Show
that the probability density function for the (central) Wishart

distribution
is proportional to

 

»

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