Consistency of the generalized likelihood ratio test for heteroscedastic normal mixtures

We study a generalized likelihood ratio test for heteroscedastic normal mixtures. The test is based on the generalized maximum likelihood estimator in the context of demixing. We prove that the test is consistent throughout the detectable region of the sparse heteroscedastic mixtures. The test requires specification of a lower bound for the variances. We study the effect of the misspecification of the lower bound. It turns out that the misspecification does not have an adverse effect on the power of the test. We demonstrate the satisfactory power of the test in detecting non-null component by comparing with several state-of-the-art procedures. Copyright (c) 2017 John Wiley & Sons, Ltd.