Suppose Z is an m-dimensional multivariate normal random variable with meam vector u and correlation matrix M, meaning that each component of Z has variance 1. Suppose the rank of M is k, less than m, then the commulative distribution function F of Z is not absolutely continuous with respect to the m-dimensional Lebesgure measure v. However, another measure w can be found such that w << F, so that with respect to w, F has a density function.

What is the density, dF/ dw? How would the likelihood ratio test for two such singular Z's with different mean vectors but the same correlation matrix look like?

Thank you.