Signal detection via Phi-divergences for general mixtures

TL;DR

This paper explores the use of Phi-divergence-based goodness-of-fit tests for detecting signals in high-dimensional noise, extending their optimality from normal mixtures to more general models.

Contribution

It introduces a technique to transfer the optimality of Phi-divergence tests to a broad class of mixture models, including exponential families.

Findings

The Phi-divergence family is optimal for dense and sparse signal models.

The tests have no power at the detection boundary, unlike the likelihood ratio test.

Application to exponential and Gumbel distributions demonstrates broad applicability.

Abstract

In this paper we are interested in testing whether there are any signals hidden in high dimensional noise data. Therefore we study the family of goodness-of-fit tests based on $\Phi$-divergences including the test of Berk and Jones as well as Tukey's higher criticism test. The optimality of this family is already known for the heterogeneous normal mixture model. We now present a technique to transfer this optimality to more general models. For illustration we apply our results to dense signal and sparse signal models including the exponential-$\chi^2$ mixture model and general exponential families as the normal, exponential and Gumbel distribution. Beside the optimality of the whole family we discuss the power behavior on the detection boundary and show that the whole family has no power there, whereas the likelihood ratio test does.