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Semiparametric mixtures of nonparametric regressions

Author

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  • Sijia Xiang

    (Zhejiang University of Finance and Economics)

  • Weixin Yao

    (University of California)

Abstract

In this article, we propose and study a new class of semiparametric mixture of regression models, where the mixing proportions and variances are constants, but the component regression functions are smooth functions of a covariate. A one-step backfitting estimate and two EM-type algorithms have been proposed to achieve the optimal convergence rate for both the global parameters and the nonparametric regression functions. We derive the asymptotic property of the proposed estimates and show that both the proposed EM-type algorithms preserve the asymptotic ascent property. A generalized likelihood ratio test is proposed for semiparametric inferences. We prove that the test follows an asymptotic $$\chi ^2$$ χ 2 -distribution under the null hypothesis, which is independent of the nuisance parameters. A simulation study and two real data examples have been conducted to demonstrate the finite sample performance of the proposed model.

Suggested Citation

  • Sijia Xiang & Weixin Yao, 2018. "Semiparametric mixtures of nonparametric regressions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(1), pages 131-154, February.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:1:d:10.1007_s10463-016-0584-7
    DOI: 10.1007/s10463-016-0584-7
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    References listed on IDEAS

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    1. Young, D.S. & Hunter, D.R., 2010. "Mixtures of regressions with predictor-dependent mixing proportions," Computational Statistics & Data Analysis, Elsevier, vol. 54(10), pages 2253-2266, October.
    2. Matthew Stephens, 2000. "Dealing with label switching in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 795-809.
    3. Mian Huang & Weixin Yao, 2012. "Mixture of Regression Models With Varying Mixing Proportions: A Semiparametric Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 711-724, June.
    4. Wang, Shaoli & Yao, Weixin & Huang, Mian, 2014. "A note on the identifiability of nonparametric and semiparametric mixtures of GLMs," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 41-45.
    5. Goldfeld, Stephen M. & Quandt, Richard E., 1973. "A Markov model for switching regressions," Journal of Econometrics, Elsevier, vol. 1(1), pages 3-15, March.
    6. Mian Huang & Runze Li & Shaoli Wang, 2013. "Nonparametric Mixture of Regression Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 929-941, September.
    7. Zhang, Yongli & Yang, Yuhong, 2015. "Cross-validation for selecting a model selection procedure," Journal of Econometrics, Elsevier, vol. 187(1), pages 95-112.
    8. Yao, Weixin & Lindsay, Bruce G., 2009. "Bayesian Mixture Labeling by Highest Posterior Density," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 758-767.
    9. Clifford M. Hurvich & Jeffrey S. Simonoff & Chih‐Ling Tsai, 1998. "Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 271-293.
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    Cited by:

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    2. Yanyuan Ma & Shaoli Wang & Lin Xu & Weixin Yao, 2021. "Semiparametric mixture regression with unspecified error distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 429-444, June.
    3. Xue, Jiacheng & Yao, Weixin, 2022. "Machine Learning Embedded Semiparametric Mixtures of Regressions with Covariate-Varying Mixing Proportions," Econometrics and Statistics, Elsevier, vol. 22(C), pages 159-171.
    4. Marco Berrettini & Giuliano Galimberti & Saverio Ranciati, 2023. "Semiparametric finite mixture of regression models with Bayesian P-splines," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(3), pages 745-775, September.

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