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Mixture of Regression Models With Varying Mixing Proportions: A Semiparametric Approach

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  • Mian Huang
  • Weixin Yao

Abstract

In this article, we study a class of semiparametric mixtures of regression models, in which the regression functions are linear functions of the predictors, but the mixing proportions are smoothing functions of a covariate. We propose a one-step backfitting estimation procedure to achieve the optimal convergence rates for both regression parameters and the nonparametric functions of mixing proportions. We derive the asymptotic bias and variance of the one-step estimate, and further establish its asymptotic normality. A modified expectation-maximization-type (EM-type) estimation procedure is investigated. We show that the modified EM algorithms preserve the asymptotic ascent property. Numerical simulations are conducted to examine the finite sample performance of the estimation procedures. The proposed methodology is further illustrated via an analysis of a real dataset.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:498:p:711-724
    DOI: 10.1080/01621459.2012.682541
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    Citations

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    Cited by:

    1. Lu, Xiaosun & Huang, Yangxin & Zhu, Yiliang, 2016. "Finite mixture of nonlinear mixed-effects joint models in the presence of missing and mismeasured covariate, with application to AIDS studies," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 119-130.
    2. Hoshino Tadao & Yanagi Takahide, 2022. "Estimating marginal treatment effects under unobserved group heterogeneity," Journal of Causal Inference, De Gruyter, vol. 10(1), pages 197-216, January.
    3. Sijia Xiang & Weixin Yao, 2020. "Semiparametric mixtures of regressions with single-index for model based clustering," 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. 14(2), pages 261-292, June.
    4. Yen-Chi Chen, 2017. "Modal Regression using Kernel Density Estimation: a Review," Papers 1710.07004, arXiv.org, revised Dec 2017.
    5. Yao, Weixin & Wei, Yan & Yu, Chun, 2014. "Robust mixture regression using the t-distribution," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 116-127.
    6. Gustavo Alexis Sabillón & Luiz Gabriel Fernandes Cotrim & Daiane Aparecida Zuanetti, 2023. "A data-driven reversible jump for estimating a finite mixture of regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 350-369, March.
    7. Ye, Mao & Lu, Zhao-Hua & Li, Yimei & Song, Xinyuan, 2019. "Finite mixture of varying coefficient model: Estimation and component selection," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 452-474.
    8. Wang, Shaoli & Huang, Mian & Wu, Xing & Yao, Weixin, 2016. "Mixture of functional linear models and its application to CO2-GDP functional data," Computational Statistics & Data Analysis, Elsevier, vol. 97(C), pages 1-15.
    9. Sphiwe B. Skhosana & Salomon M. Millard & Frans H. J. Kanfer, 2023. "A Novel EM-Type Algorithm to Estimate Semi-Parametric Mixtures of Partially Linear Models," Mathematics, MDPI, vol. 11(5), pages 1-20, February.
    10. 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.
    11. 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.
    12. Yuzhu Tian & Manlai Tang & Maozai Tian, 2016. "A class of finite mixture of quantile regressions with its applications," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(7), pages 1240-1252, July.
    13. 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.
    14. Tran, Kien C. & Tsionas, Mike G., 2016. "Zero-inefficiency stochastic frontier models with varying mixing proportion: A semiparametric approach," European Journal of Operational Research, Elsevier, vol. 249(3), pages 1113-1123.
    15. 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.
    16. 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.
    17. Kien C. Tran & Mike G. Tsionas & Emmanuel Mamatzakis, 2020. "Why fully efficient banks matter? A nonparametric stochastic frontier approach in the presence of fully efficient banks," Empirical Economics, Springer, vol. 58(6), pages 2733-2760, June.

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