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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.

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$F$ is supported on some $k$-dimensional subspace of $\mathbb{R}^m$, and by diagonalizing $M$ you can see what subspace, and how to get $Z$ by a linear transformation of a standard $k$-dimensional multivariate normal (covariance matrix $I$). So the support of $w$ must be the same subspace, and everything reduces to the non-singular $k$-dimensional case. This is not a research level question and I am voting to close. –  Nate Eldredge Apr 14 '11 at 18:27
    
Unless the two densities have the same $m-k$-dimensional affine space as support, the likelihood ratio is undefined with probability 1. –  Hans Engler Apr 14 '11 at 19:14
    
Thank you both for pointing them out and they are what I am struggling with. It is generally known that F is supported on the k-dim linear subspace G spanned by Q in the Cholesky decomposition of M. But what is the analytic form of this density, dF/dw? Let us say, Z1 ~ Normal(u1, M), Z2~ Normal(u2,M). Then Z1-u1, Z2-u2 are in the same space G and the likelihood ratio is well-defined for the translated Z1 and Z2, right? Regards, -Chee –  Chee Apr 17 '11 at 0:58
    
Dear All, This is a research level question! But it has been ignored simply because people just know Z is in a rank(M)-dimensional subspace which generates a measure that is induced by the probabilty measure generated by a rank(M)-dim random vector and few are interested or ever have to work with singular densities. There is a link found in a 1968 paper but I am not sure whether I have correctly interpreted it. There is an analytic form for this dF/dw, but it is not unique! I appreciate any insights provided by you! Thanks, -Chee –  Chee Apr 23 '11 at 0:30
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