Signal Detection under Composite Hypotheses with Identical Distributions for Signals and for Noises

Abstract

In this paper, we consider the problem of detecting signals in multiple, sequentially observed data streams, where the distribution of each stream lies in one of two common composite spaces, depending on whether it is a signal or a noise. For this problem, we study a practical yet underexplored setting where it is a priori known that all signals have an identical distribution and so do all noises. Compared to the general setting where local distributions are free to take any values, this structure facilitates faster decision-making thanks to a smaller joint distribution space. However, it introduces additional challenges to the analysis of problem and design of tests, since the local distributions are now coupled. In this paper, we first establish a universal lower bound on the minimum expected sample size, which characterizes the essential difficulty of the problem and involves constants that are neither the minimum Kullback-Leibler divergences between the signal/noise distribution to the noise/signal distribution space, which appear in the lower bound for the general setting, nor the Kullback-Leibler divergences between the signal distribution and the noise distribution. Besides, we propose a test that controls the two types of familywise error rates below arbitrary levels, and achieves the minimum expected sample size asymptotically as the levels go to zero. Numerical studies are presented to compare with the state-of-the-art test for the general setting and demonstrate robustness against model misspecification.