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

TL;DR

This paper addresses the challenge of detecting signals in multiple data streams with unknown but structured distributions, proposing a new testing procedure that is both error-controlled and asymptotically optimal in sample size.

Contribution

It introduces a universal lower bound and a novel testing method for signal detection with shared parameters, improving decision speed and error control.

Findings

Proposed a universal lower bound on expected sample size.

Developed a testing procedure controlling familywise error probabilities.

Achieved asymptotic optimality in sample size under all distributions.

Abstract

In this paper, we consider the problem of detecting signals in multiple, sequentially observed data streams. For each stream, the exact distribution is unknown, but characterized by a parameter that takes values in either of two disjoint composite spaces depending on whether it is a signal or noise. Furthermore, we consider a practical yet underexplored setting where all signals share the same parameter, as do all noises. Compared to the unconstrained case where the parameters in all streams are allowed to vary, this assumption facilitates faster decisionmaking thanks to the smaller parameter space. However, it introduces additional challenges in the analysis of the problem and designing of testing procedures since the local parameters are now coupled. In this paper, we establish a universal lower bound on the minimum expected sample size that characterizes the inherent difficulty of the problem. Moreover, we propose a novel testing procedure which not only controls the familywise error probabilities below arbitrary, user-specified levels, but also achieves the minimum expected sample size under every possible distribution asymptotically, as the levels go to zero.