Electronic Journal of Statistics

Detectability of nonparametric signals: higher criticism versus likelihood ratio

Marc Ditzhaus and Arnold Janssen

Full-text: Open access

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 behaviour on the detection boundary by discussing, among others, Pitmans’s asymptotic efficiency as an application of Le Cam’s theory.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 4094-4137.

Dates
Received: June 2018
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1544670253

Digital Object Identifier
doi:10.1214/18-EJS1502

Mathematical Reviews number (MathSciNet)
MR3890763

Zentralblatt MATH identifier
07003238

Subjects
Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 62G32: Statistics of extreme values; tail inference

Keywords
Nonparametric sparse signals infinitely divisible distribution detection boundary and regions Tukey’s higher criticism Le Cam’s local asymptotic normality

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ditzhaus, Marc; Janssen, Arnold. Detectability of nonparametric signals: higher criticism versus likelihood ratio. Electron. J. Statist. 12 (2018), no. 2, 4094--4137. doi:10.1214/18-EJS1502. https://projecteuclid.org/euclid.ejs/1544670253


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