Statistics Papers

Document Type

Journal Article

Date of this Version

11-2011

Publication Source

Journal of the Royal Statistical Society: Series B

Volume

73

Issue

5

Start Page

629

Last Page

662

DOI

10.1111/j.1467-9868.2011.00778.x

Abstract

The problem of detecting heterogeneous and heteroscedastic Gaussian mixtures is considered. The focus is on how the parameters of heterogeneity, heteroscedasticity and proportion of non-null component influence the difficulty of the problem. We establish an explicit detection boundary which separates the detectable region where the likelihood ratio test is shown to detect the presence of non-null effects reliably from the undetectable region where no method can do so. In particular, the results show that the detection boundary changes dramatically when the proportion of non-null component shifts from the sparse regime to the dense regime. Furthermore, it is shown that the higher criticism test, which does not require specific information on model parameters, is optimally adaptive to the unknown degrees of heterogeneity and heteroscedasticity in both the sparse and the dense cases.

Copyright/Permission Statement

This is the peer reviewed version of the following article: Tony Cai, T., Jessie Jeng, X. and Jin, J. (2011), Optimal detection of heterogeneous and heteroscedastic mixtures. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73: 629–662., which has been published in final form at doi: 10.1111/j.1467-9868.2011.00778.x. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving http://olabout.wiley.com/WileyCDA/Section/id-820227.html#terms.

Keywords

detection boundary, higher criticism, likelihood ratio test, optimal adaptivity, sparsity

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Date Posted: 27 November 2017

This document has been peer reviewed.