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The Consistency of Estimators in Finite Mixture Models

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  • R. C. H. Cheng
  • W. B. Liu

Abstract

The parameters of a finite mixture model cannot be consistently estimated when the data come from an embedded distribution with fewer components than that being fitted, because the distribution is represented by a subset in the parameter space, and not by a single point. Feng & McCulloch (1996) give conditions, not easily verified, under which the maximum likelihood (ML) estimator will converge to an arbitrary point in this subset. We show that the conditions can be considerably weakened. Even though embedded distributions may not be uniquely represented in the parameter space, estimators of quantities of interest, like the mean or variance of the distribution, may nevertheless actually be consistent in the conventional sense. We give an example of some practical interest where the ML estimators are root of n‐consistent. Similarly consistent statistics can usually be found to test for a simpler model vs a full model. We suggest a test statistic suitable for a general class of model and propose a parameter‐based bootstrap test, based on this statistic, for when the simpler model is correct.

Suggested Citation

  • R. C. H. Cheng & W. B. Liu, 2001. "The Consistency of Estimators in Finite Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(4), pages 603-616, December.
    • Handle: RePEc:bla:scjsta:v:28:y:2001:i:4:p:603-616
    DOI: 10.1111/1467-9469.00257
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    Cited by:

    1. F. Bartolucci & A. Farcomeni & F. Pennoni, 2014. "Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 433-465, September.
    2. Yoichi Miyata & Takayuki Shiohama & Toshihiro Abe, 2020. "Estimation of finite mixture models of skew-symmetric circular distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(8), pages 895-922, November.
    3. Tin Lok James Ng & Thomas Brendan Murphy, 2021. "Model-based Clustering of Count Processes," Journal of Classification, Springer;The Classification Society, vol. 38(2), pages 188-211, July.
    4. Ahfock, Daniel & McLachlan, Geoffrey J., 2021. "Harmless label noise and informative soft-labels in supervised classification," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    5. Pennoni, Fulvia & Romeo, Isabella, 2016. "Latent Markov and growth mixture models for ordinal individual responses with covariates: a comparison," MPRA Paper 72939, University Library of Munich, Germany.
    6. Yoshikazu Terada, 2014. "Strong Consistency of Reduced K-means Clustering," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 913-931, December.

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