Asymptotic Karlin-Rubin’s Theorem with Application to Signal Detection in a Subspace Cone

We first propose an asymptotic formulation of Karlin-Rubin’s theorem that relies on the weak convergence of a sequence of random vectors to design Asymptotically Uniformly Most Powerful (AUMP) tests dedicated to composite hypotheses. This general property of optimality is then applied to the problem of testing whether the energy of a signal projected onto a known subspace exceeds a specified proportion of its total energy. The signal is assumed unknown deterministic and it is observed in independent and additive white Gaussian noise. Such a problem can arise when the signal to be detected obeys the linear subspace model and when it is corrupted by unknown interference. It can also be relevant in machine learning applications where one wants to check whether an assumed linear model fits the analyzed data. For this problem, where it is shown that no Uniformly Most Powerful (UMP) and no UMP invariant tests exist, an AUMP invariant test is derived.