4.1 The Multivariate Normal Distribution

Let U = ( U ₁, U ₂, , U _n) be a vector of independent standard normal variables of length n and let A be an n n non-singular matrix. Furthermore, let = ( ₁, ₂, , _n) be a vector of constants of length n. Let

Then X is said to be a vector of multinormal variables. Following the method of Section 3.10.2, we find

since E( UU ^T) = I, where I is the n n unit matrix.

Conversely, let X be a multivariate normal vector with mean .

Let ? = E[( X ? )( X ? ) ^T] be the (non-singular) variance covariance matrix of X.

Suppose that we can find a decomposition of ? in the form

where A is a non-singular square matrix. Then U = A ^{? 1}( X ? ) will be a vector of independent standard normal variables. For a proof see Ref. [40, p. 347]. This is called a process of standardization of X.

One well-known form of the decomposition (4.5) is the Choleski decomposition, where A is chosen to be a lower triangular matrix. For details see Ref. [63]. Most computer programs have an implementation of the Choleski algorithm.

It is...