Hypothesis Testing in Mixture Regression Models

Abstract

We establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models. Moreover, under specific and reasonable conditions, we show that the optimal convergence rate of "n"-super- - 1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators. We also derive the asymptotic distributions of two log-likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the "p"-value. Simulation studies are conducted to investigate the empirical performance of the two test statistics. Finally, two real data sets are analysed to illustrate the application of our theoretical results. Copyright 2004 Royal Statistical Society.

Testing the Number of Components in Normal Mixture Regression Models

“…Despite its importance, testing for the number of components in normal mixture regression models has been a long-standing unsolved problem because the standard asymptotic analysis of the likelihood ratio test (LRT) statistic breaks down due to problems such as non-identifiable parameters and the true parameter being on the boundary of the parameter space. Numerous papers have been written on the subject of the likelihood ratio test for the number of components (see, e.g., Ghosh and Sen, 1985;Chernoff and Lander, 1995;Lemdani and Pons, 1997;Chen andChen, 2001, 2003;Chen et al, 2004;Garel, 2001Garel, , 2005, and the asymptotic distribution of the LRT statistic for general finite mixture models has been derived as a functional of the Gaussian process (Dacunha-Castelle and Gassiat, 1999;Azaïs et al, 2009;Liu and Shao, 2003;Zhu and Zhang, 2004).…”

“…This leads to the loss of "strong identifiability" condition introduced by Chen (1995). As a result, neither Assumption (P1) of Dacunha-Castelle and Gassiat (1999) nor Assumption 7 of Azaïs et al (2009) holds, and Assumption 3 of Zhu and Zhang (2004) is violated, while Corollary 4.1 of Liu and Shao (2003) does not hold in heteroscedastic normal mixtures.…”

Competition in Lending: Theory and Experiments*

“…The distribution of the statistic is presented in Zhu and Zhang (2004). The mixture model maximum log likelihood estimates are based on the EM algorithm, which is standard for estimating mixture models.…”

Robust Subgroup Identification

“…However, most mixture-model-based approaches require specifying the number of components, and the underlying distribution for each component, for which the most popular choice is normal distribution giving rise to normal mixture models. It's well known that testing for the number of components in mixture models is technically challenging due to the nonidentifiability of parameters under the null hypothesis; see Zhu and Zhang (2004), Chen and Li (2009), Li and Chen (2010), Kasahara and Shimotsu (2015), Shen and He (2015) for some related discussions. On the other hand, the normality assumption for normal mixture models may be restrictive or susceptible to outliers.…”