Hypothesis Testing in Mixture Regression Models

Zhu, Hong; Zhang, Heping

doi:10.1046/j.1369-7412.2003.05379.x

Cited by 69 publications

(54 citation statements)

References 26 publications

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“…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).…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Testing the Number of Components in Normal Mixture Regression Models

Kasahara

Shimotsu

2015

Journal of the American Statistical Association

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Testing the number of components in finite normal mixture models is a longstanding challenge because of its non-regularity. This paper studies likelihood-based testing of the number of components in normal mixture regression models with heteroscedastic components. We construct a likelihood-based test of the null hypothesis of m 0 components against the alternative hypothesis of m 0 + 1 components for any m 0 . The null asymptotic distribution of the proposed modified EM test statistic is the maximum of m 0 random variables that can be easily simulated. The simulations show that the proposed test has very good finite sample size and power properties.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Testing the Number of Components in Normal Mixture Regression Models

Kasahara

Shimotsu

2015

Journal of the American Statistical Association

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“…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.…”

Section: Baseline Environmentmentioning

confidence: 99%

Competition in Lending: Theory and Experiments*

Asparouhova

2006

Review of Finance

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A variation of the Rothschild-Stiglitz' equilibrium is examined in the context of competitive lending under adverse selection. The predictions of the model are tested in an experimental market setting. If equilibrium exists, it predicts that the loan contracts offered and taken separate the projects being financed by quality. When equilibrium exists, the experiments confirm the theory.The entrepreneurs with high-risk projects take bigger loans and bear higher credit spreads than those with low-risk projects. When equilibrium does not exist, which happens exactly when the candidate for equilibrium does not provide a Pareto-optimal allocation, in half of the sessions loan trading stabilizes around the candidate equilibrium pair. In the other half, however, markets never settle down. This finding has important implications. When lenders can offer menus of contracts, as is usually the case in the real world, the outcome may not be the zero-profit separating contracts of the standard model in the theory of corporate finance. Worse, as an example in the paper illustrates, fitting the standard model to field data may lead to serious biases in estimated parameters while falsely accepting the model's main restriction (separation). * The financial support of the Division of Humanities and Social Sciences at Caltech is gratefully acknowledged. I would like to thank Charles Plott, Thomas Palfrey, Bill Zame, Mike Lemmon, as well as the seminar participants at Caltech, UCSD, Duke, Berkeley, Stanford, University of Utah, Columbia, Georgia State University, Tulane, University of Houston, and Arizona State University for helpful comments. I am especially grateful to Peter Bossaerts for all his suggestions and comments, and the staff of EEPS, SSEL, and CASSEL for their help in running the experiments. All errors are my own.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Robust Subgroup Identification

Zhang¹,

Wang²,

Zhu³

2019

STAT SINICA

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In many applications, subgroups with different parameters may exist even after accounting for the covariate effects, and it is important to identify the meaningful subgroups for better medical treatment or market segmentation. We propose a robust subgroup identification method based on median regression with concave fusion penalizations.The proposed method can simultaneously determine the number of subgroups, identify the group membership for each subject, and estimate the regression coefficients. Without requiring any parametric distributional assumptions, the proposed method is robust against outliers in the response and heteroscedasticity in the regression error. We develop a convenient algorithm based on local linear approximation, and establish the oracle property of the proposed penalized estimator and the model selection consistency for the modified Bayesian information criteria. The numerical performance of the proposed method is assessed through simulation and the analysis of a heart disease data.

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Hypothesis Testing in Mixture Regression Models

Cited by 69 publications

References 26 publications

Testing the Number of Components in Normal Mixture Regression Models

Testing the Number of Components in Normal Mixture Regression Models

Competition in Lending: Theory and Experiments*

Robust Subgroup Identification

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