Asymptotics for the likelihood ratio test in a two-component normal mixture model

Abstract

This paper characterizes the asymptotic properties of the likelihood ratio test (LRT) statistic for testing homogeneity in a two-component normal mean mixture model. We justify that the LRT statistic 2 λ n is asymptotically equivalent to the square of the supremum of the stochastic process studied in Bickel and Chernoff (Statistics and Probability: A Raghu Raj Bahadur Festschrift (1993) 83). In particular, we prove that 2 λ n diverges to +∞ at a rate of log log n which confirms a conjecture of Hartigan (Proceedings of Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (1985)). More specifically, under the null hypothesis we prove the following fact: lim n→∞ P {2λ n− log log n+ log(2π 2)⩽x}= exp(− e −x/2), x∈ R.