Suppose I observe i.i.d. $x_i \sim \mathcal{N}\left(\mu,\Sigma\right)$, and wish to test $H_0: A\ $vech$\left(\Sigma^{-1}\right) = a$ for a conformable matrix $A$ and vector $a$. Is there known work on this problem?

The obvious (to me) attempt would be via a likelihood ratio test, but it seems like maximizing the likelihood subject to the constraints of $H_0$ would require a SDP solver and could be pretty hairy.

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    $\begingroup$ Do you have any additional constraints on $A$? If $A$ is invertible, then $H_0=\rm{vech}(\Sigma^{-1})=A^{-1}a$. The then problem amounts to a well-known problem: that of testing whether $\Sigma^{-1}=B$. Here $\rm{vech}(B)=A^{-1}a$ (remember that $\rm{vech}(B)$ determines $B$ uniquely). $\endgroup$ – MånsT Jun 18 '12 at 9:49
  • $\begingroup$ @MånsT ; sadly I am interested in the general case. Typically $A$ will have around 10 rows and 400 columns or so. $\endgroup$ – shabbychef Jun 18 '12 at 16:50
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    $\begingroup$ One thing I am wondering about this problem regards feasibility. Obviously it is easy to find pairs $(A,a)$ such that no positive semidefinite matrix could satisfy the constraints. Potentially more troublesome for a likelihood ratio test is it would seem there can be instances in which even when the null hypothesis were true, with high probability one obtains an infeasible problem instance. Perhaps that last part is mistaken though. (+1) You tend to ask interesting and challenging problems. I enjoy reading and thinking a bit about them. $\endgroup$ – cardinal Jun 23 '12 at 17:25
  • $\begingroup$ @cardinal Good catch! I had not thought of that because, in the application I am considering, the null hypothesis restricts only non-diagonal elements of $\Sigma^{-1}$ (the corresponding columns of $A$ are all zero). Since the diagonal can be arbitrarily large, I can guarantee feasibility. $\endgroup$ – shabbychef Jun 24 '12 at 4:06

Beran and Srivastava (1985, Annals of Statistics) had a paper where they proposed a general bootstrap approach to apply a rotation to the covariance matrix that make it match the distribution under the null. @cardinal's point about existence of such a matrix is highly relevant here though. You need to be able to come up with at least some sort of approximation for a matrix that satisfies the constraints you impose under the null.

Chen, Variyath and Bovas had a paper on adjusted empirical likleihood where they demonstrated how it can be used to test a rather weird structure on the covariance matrix. I think this paper eventually came out in CJS.

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  • $\begingroup$ I'm not sure I can easily translate these into a solution to my problem, but they are both fascinating reads. +1. $\endgroup$ – shabbychef Jun 26 '12 at 18:01

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