Theory of the Multinormal

In the preceeding chapter we realized the importance of the multivariate normal distribution, its geometry and connection with elliptic dependence structures. The multivariate normal comes into play in many applications and statistical tests. It is therefore important to know how this distribution behaves when we apply conditioning or linear or nonlinear transformation. It is also of interest to check whether partitioned random vectors are still normally distributed and how the multinormal distribution is popping out of theoretical concepts. 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. We consider here best linear approximations, partial correlation (expressed via partitioned matrices), and conditioning on parts of a multinormal random vector.