Suppose that
is a standard normal random vector, i.e.,
, independent of the random matrix
.
What is the distribution of
? The answer is
provided by the Hotelling
-distribution:
is Hotelling
distributed.
The Hotelling
-distribution
is a generalization of the Student
-distribution.
The general multinormal distribution
is considered
in Theorem 5.8. The Hotelling
-distribution will play
a central role in hypothesis testing in Chapter 7.
COROLLARY 5.3
If
![$\overline x$](mvahtmlimg1020.gif)
is the mean
of a sample drawn from a normal population
![$N_p(\mu,\Sigma)$](mvahtmlimg1282.gif)
and
![$\data{S}$](mvahtmlimg687.gif)
is the sample covariance matrix, then
![\begin{displaymath}
(n-1)(\overline x-\mu )^{\top} \data{S}^{-1}(\overline x-\mu...
...-\mu)^{\top} \data{S}^{-1}_u(\overline x-\mu )\sim T^2(p,n-1).
\end{displaymath}](mvahtmlimg1764.gif) |
(5.17) |
Recall that
is an unbiased
estimator of the covariance matrix.
A connection between the Hotelling
- and the
-distribution
is given by the next theorem.
EXAMPLE 5.5
In the univariate case (p=1), this theorem boils down to the well
known result:
For further details on Hotelling
-distribution see
Mardia et al. (1979).
The next corollary follows immediately from (3.23),(3.24)
and from Theorem 5.8. It will be useful for testing linear restrictions
in multinormal populations.
COROLLARY 5.4
Consider a linear transform of
![$X \sim N_p ( \mu,\Sigma),\ Y = \data{A}X$](mvahtmlimg1768.gif)
where
![$\data{A}(q\times p)$](mvahtmlimg1644.gif)
with
![$(q \leq p).$](mvahtmlimg1769.gif)
If
![$\overline x$](mvahtmlimg1020.gif)
and
![$\data{S}_X$](mvahtmlimg1770.gif)
are the sample mean
and the covariance matrix, we have
The
distribution is closely connected to the univariate
-statistic.
In Example 5.4 we described the manner in which the Wishart
distribution generalizes the
-distribution.
We can write (5.17) as:
which is of the form
This is analogous to
or
for the univariate case. Since the multivariate normal and Wishart random
variables are independently distributed, their joint distribution is the
product of the marginal normal and Wishart distributions. Using calculus, the
distribution of
as given above can be derived from
this joint distribution.
Summary
![$\ast$](mvahtmlimg108.gif)
-
Hotelling's
-distribution is a generalization of the
-distribution. In particular
.
![$\ast$](mvahtmlimg108.gif)
-
has a
distribution.
![$\ast$](mvahtmlimg108.gif)
-
The relation between Hotelling's
and Fisher's
-distribution is given by