A Selective Overview of Semiparametric Mixture of Regression Models

  • Sijia XiangEmail author
  • Weixin Yao
Part of the ICSA Book Series in Statistics book series (ICSABSS)


Finite mixture of regression models have been popularly used in many applications. In this article, we did a systematic review of newly developed semiparametric mixture of regression models. Recent developments and some open questions are also discussed.



Xiang’s research is supported by Zhejiang Provincial NSF of China [grant no. LQ16A010002] and NSF of China [grant no. 11601477]. Yao’s research is supported by NSF [grant no. DMS-1461677] and Department of Energy with the award DE-EE0007328.


  1. Bordes, L., Kojadinovic, I., & Vandekerkhove, P. (2013). Semiparametric estimation of a two-component mixture of linear regressions in which one component known. Electronic Journal of Statistics, 7, 2603–2644.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Cao, J., & Yao, W. (2012). Semiparametric mixture of binomial regression with a degenerate component. Statistica Sinica, 22, 27–46.MathSciNetzbMATHGoogle Scholar
  3. Chen, J., & Tan, X. (2009). Inference for multivariate normal mixtures. Journal of Multivariate Analysis, 100, 1367–1383.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Chen, J., Tan, X., & Zhang, R. (2008). Inference for normal mixture in mean and variance. Statistica Sincia, 18, 443–465.MathSciNetzbMATHGoogle Scholar
  5. Dziak, J. J., Li, R., Tan, X., Shiffman, S., & Shiyko, M. P. (2015). Modeling intensive longitudinal data with mixtures of nonparametric trajectories and time-varying effects. Psychological Methods, 20(4), 444–469.CrossRefGoogle Scholar
  6. Faicel, C. (2016). Unsupervised learning of regression mixture models with unknown number of components. Journal of Statistical Computation and Simulation, 86(12), 2308–2334.MathSciNetCrossRefGoogle Scholar
  7. Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications. London: Chapman & Hall.zbMATHGoogle Scholar
  8. Frühwirth-Schnatter, S. (2001). Markov chain monte carlo estimation of classical and dynamic switching and mixture models. Journal of American and Statistical Association, 96, 194–209.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Frühwirth-Schnatter, S. (2006). Finite mixture and Markov switching models. New York: Springer.Google Scholar
  10. Goldfeld, S. M., & Quandt, R. E. (1973). A Markov model for switching regression. Journal of Econometrics, 1, 3–15.zbMATHCrossRefGoogle Scholar
  11. Green, P. J., & Richardson, S. (2002). Hidden Markov models and disease mapping. Journal of American and Statistical Association, 97, 1055–1070.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Härdle, W., Hall, P., & Ichimura, H. (1993). Optimal smoothing in single-index models. Annals of Statistics, 21, 157–178.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Hathaway, R. J. (1985). A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Annals of Statistics, 13, 795–800.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Hathaway, R. J. (1986). A constrained EM algorithm for univariate mixtures. Journal of Statistical Computation and Simulation, 23, 211–230.CrossRefGoogle Scholar
  15. Hu, H., Yao, W., & Wu, Y. (2017). The robust EM-type algorithms for log-concave mixtures of regression models. Computational Statistics & Data Analysis, 111, 14–26.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Huang, M., & Yao, W. (2012). Mixture of regression models with varying mixing proportions: a semiparametric approach. Journal of the American Statistical Association, 107(498), 711–724.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Huang, M., Li, R., & Wang, S. (2013). Nonparametric mixture of regression models. Journal of the American Statistical Association, 108(503), 929–941.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Huang, M., Li, R., Wang, H., & Yao, W. (2014). Estimating mixture of Gaussian processes by kernel smoothing. Journal of Business & Economic Statistics, 32(2), 259–270.MathSciNetCrossRefGoogle Scholar
  19. Huang, M., Ji, Q., & Yao, W. (2017). Semiparametric hidden Markov model with nonparametric regression. Communications in Statistics-Theory and Methods. Scholar
  20. Huang, M., Yao, W., Wang, S., & Chen, Y. (2018). Statistical inference and application of mixture of varying coefficient models. Scandinavian Journal of Statistical models, 45(3), 618–643.zbMATHCrossRefGoogle Scholar
  21. Hunter, D. R., & Young, D. S. (2012). Semiparametric mixtures of regressions. Journal of Nonparametric Statistics, 24(1), 19–38.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics, 58, 71–120.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Jacobs, R. A., Jordan, M. I., Nowlan, S. J., & Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation, 3(1), 79–87.CrossRefGoogle Scholar
  24. Montuelle, L., & Le Pennec, E. (2014). Mixture of Gaussian regressions model with logistic weights, a penalized maximum likelihood approach. Electronic Journal of Statistics, 8, 1661–1695.MathSciNetzbMATHCrossRefGoogle Scholar
  25. Pena, D., Rodríguez, J., & Tiao, G. C. (2003). Identifying mixtures of regression equations by the SAR procedure. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, & M. West (Eds.), Bayesian statistics (Vol. 7, pp. 327–348). Oxford: Clarendon Press.Google Scholar
  26. Sapatnekar, S. S. (2011). Overcoming variations in nanometer-scale technologies. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 1(1), 5–18.CrossRefGoogle Scholar
  27. Tan, X., Shiyko, M. P., Li, R., Li, Y., & Dierker, L. (2012). A time-varying effect model for intensive longitudinal data. Psychological Methods, 17(1), 61–77.CrossRefGoogle Scholar
  28. Vandekerkhove, P. (2013). Estimation of a semiparametric mixture of regressions model. Journal of Nonparametric Statistics, 25, 181–208.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Viele, K., & Tong, B. (2002). Modeling with mixtures of linear regressions. Statistics and Computing, 12, 315–330.MathSciNetCrossRefGoogle Scholar
  30. Wang, S., Yao, W., & Huang, M. (2014). A note on the identifiability of nonparametric and semiparametric mixtures of GLMs. Statistics and Probability Letters, 93, 41–45.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Wang, S., Huang, M., Wu, X., & Yao, W. (2016). Mixture of functional linear models and its application to CO2-GDP functional data. Computational Statistics & Data Analysis, 97, 1–15.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Wedel, M., & DeSarbo, W. S. (1993). A latent class binomial logit methodology for the analysis of paired comparison data. Decision Sciences, 24, 1157–1170.CrossRefGoogle Scholar
  33. Wu, Q., & Yao, W. (2016). Mixtures of quantile regressions. Computational Statistics & Data Analysis, 93, 162–176.MathSciNetzbMATHCrossRefGoogle Scholar
  34. Xiang, S., & Yao, W. (2016). Semiparametric mixtures of nonparametric regressions. Annals of the Institute of Statistical Mathematics. Scholar
  35. Xiang, S., & Yao, W. (2017). Semiparametric mixtures of regressions with single-index for model based clustering. arXiv:1708.04142v1.Google Scholar
  36. Yao, W. (2010). A profile likelihood method for normal mixture with unequal variance. Journal of Statistical Planning and Inference, 140, 2089–2098.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Yao, F., Fu, Y., & Lee, T. C. M. (2011). Functional mixture regression. Biostatistics, 12, 341–353.CrossRefGoogle Scholar
  38. Young, D. S. (2014). Mixtures of regressions with changepoints. Statistical Computations, 24, 265–281.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Young, D. S., & Hunter, D. R. (2010). Mixtures of regressions with predictor-dependent mixing proportions. Computational Statistics & Data Analysis, 54, 2253–2266.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Data SciencesZhejiang University of Finance & EconomicsHangzhouChina
  2. 2.Department of StatisticsUniversity of CaliforniaRiversideUSA

Personalised recommendations