The Wishart distribution (named after its discoverer) plays a prominent role
in the analysis of estimated covariance
matrices.
If the mean of
is known to be
, then
for a data matrix
the estimated covariance matrix is
proportional to
.
This is the point where the Wishart distribution comes in, because
has a Wishart distribution
.
When we talk about the distribution of a matrix, we mean of course the
joint distribution of all its elements. More exactly: since
is symmetric we only need to consider the elements of
the lower triangular matrix
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(5.14) |
![]() |
(5.15) |
Linear transformations of the data matrix also lead to Wishart
matrices.
With this theorem we can standardize Wishart matrices since
with
the
distribution of
is
.
Another connection to the
-distribution is given by
the following theorem.
This theorem is an immediate consequence of Theorem 5.5 if we apply
the linear transformation
.
Central to the analysis of covariance matrices is the next theorem.
The following properties are useful:
For further details on the Wishart distribution see Mardia et al. (1979).