5. Theory of the Multinormal
In the preceeding chapter we saw how the multivariate normal distribution
comes into play in many applications. It is useful to know more about this
distribution, since it is often a good approximate distribution in
many situations. Another reason for considering the multinormal
distribution relies on the fact that it has many appealing properties:
it is stable under linear transforms, zero correlation corresponds to
independence, the marginals and all the conditionals are also multivariate
normal variates, etc.
The mathematical properties of the multinormal make
analyses much simpler.
In this chapter we will first concentrate
on the probabilistic properties of the
multinormal, then we will introduce two ``companion'' distributions of
the multinormal which naturally appear when sampling from a
multivariate normal population: the
Wishart and the Hotelling distributions. The latter is particularly important
for most of the testing procedures proposed in Chapter 7.