Detectability of nonparametric signals: higher criticism versus likelihood ratio

TL;DR

This paper compares the effectiveness of higher criticism and likelihood ratio tests in detecting rare, weak signals in high-dimensional noise, extending results to nonparametric models and analyzing behavior at the detection boundary.

Contribution

It introduces a new technique to prove the equivalence of HC and LLR tests in general models and provides a complete analysis of HC's performance at the detection boundary.

Findings

HC achieves the same detection region as LLR in general models.

HC remains optimal at the detection boundary for heteroscedastic normal mixtures.

New insights into LLR behavior and Pitman's efficiency at the detection boundary.

Abstract

We study the signal detection problem in high dimensional noise data (possibly) containing rare and weak signals. Log-likelihood ratio (LLR) tests depend on unknown parameters, but they are needed to judge the quality of detection tests since they determine the detection regions. The popular Tukey's higher criticism (HC) test was shown to achieve the same completely detectable region as the LLR test does for different (mainly) parametric models. We present a novel technique to prove this result for very general signal models, including even nonparametric $p$-value models. Moreover, we address the following questions which are still pending since the initial paper of Donoho and Jin: What happens on the border of the completely detectable region, the so-called detection boundary? Does HC keep its optimality there? In particular, we give a complete answer for the heteroscedastic normal mixture model. As a byproduct, we give some new insights about the LLR test's behavior on the detection boundary by discussing, among others, Pitmans's asymptotic efficiency as an application of Le Cam's theory.