GLRT Inconclusive for Gaussian Detection

Question

When constructing a detection test between two Gaussian vectors, I have $H_0: \mathcal{N}(0, \sigma_n^2)$ and $H_1: \mathcal{N}(0,\sigma_n^2 + \sigma_s^2)$, and $\sigma_n^2$ is unknown under finite samples collected, while $\sigma_s^2$ in known. Since both hypotheses are composite, I construct the GLRT while looking for ML estimate of $\sigma_n^2$ under both hypothesis. Unfortunately, I come up with both $f_0(\widehat{\sigma_w^2})$ and $f_1(\widehat{\sigma_w^2})$ to be equal and hence the GLRT is inconclusive. Am I making a mistake, or is there another way of constructing a detection test here?

Answer 1

While I'm not quite sure of the exact model you have in mind, I found strange that you say that the test is inconclusive, because—although asymptotic—the statistic has a well defined distribution: we have $$-2\log \lambda\xrightarrow{\;\mathcal{D}\;}\chi^2_r,$$ where $r$ is the difference between the number of free parameters in $H_1$ and $H_0$.

So you will always have a conclusion, be it rejecting $H_0$ or not.

But it's looking at that distribution that I saw the problem. If $\sigma^2_n$ is unknown and $\sigma^2_s$ is known, then you have one free parameter in both $H_0$ and $H_1$ with leaves you with $r=0$, that is, the GLRT does not apply. Might it be the case that it is the opposite case? Or maybe are both unknown parameters?