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

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 2λn 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 2λn diverges to +∞ at a rate of loglogn 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: limn→∞P{2λn−loglogn+log(2π2)⩽x}=exp(−e−x/2),x∈R.

[1]  Bernard Garel,et al.  Likelihood ratio test for univariate Gaussian mixture , 2001 .

[2]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[3]  B. Lindsay.  Mixture models : theory, geometry, and applications , 1995 .

[4]  Y. Shao,et al.  Asymptotics for likelihood ratio tests under loss of identifiability , 2003 .

[5]  B. Lindsay,et al.  The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family , 1994 .

[6]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[7]  N. Mendell,et al.  Simulated percentage points for the null distribution of the likelihood ratio test for a mixture of two normals. , 1988, Biometrics.

[8]  Geoffrey J. McLachlan,et al.  Mixture models : inference and applications to clustering , 1989 .

[9]  P. Révész,et al.  Strong approximations in probability and statistics , 1981 .

[10]  Jiahua Chen,et al.  Large sample distribution of the likelihood ratio test for normal mixtures , 2001 .

[11]  D. Mason,et al.  Weighted Empirical and Quantile Processes , 1986 .

[12]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[13]  R. Davies.  Hypothesis testing when a nuisance parameter is present only under the alternative , 1977 .

[14]  Cristian Pasarica,et al.  Testing Homogeneity in Gamma Mixture Models , 2003 .

[15]  P. Bickel.  Asymptotic distribution of the likelihood ratio statistic in a prototypical non regular problem , 1993 .

[16]  D. Pollard.  Asymptotics via Empirical Processes , 1989 .