Generalized Likelihood Ratio Test for Detecting Nonzero Normal Means

• Activity Number: 119
• Type: Topic Contributed
• Date/Time: Monday, August 10, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Wenhua Jiang* and Cun-Hui Zhang
• Companies: Soochow University and Rutgers University
• Keywords: sparse normal means ; generalized likelihood ratio test ; detection boundary ; multiple testing ; adaptivity

Abstract

We consider a generalized likelihood ratio test (GLRT) for testing if there is any nonzero component in a vector of normal means at all. The test is based on the generalized maximum likelihood estimator (Kiefer and Wolfowitz, 1956). We show that under the global null, that is, when all observations are i.i.d. standard normal, the order of alpha critical value of the test does not exceed (log n)^2. On the other hand, if some of normal means are nonzero, the GLRT could goes to infinity at a rate faster than (log n)^2, provided that the order of the Hellinger distance between the mixture densities under the corresponding null and alternative is larger than (log n)^2/n. We demonstrate the power of the proposed test for moderate samples by numerical experiments.