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Strong Consistency of the Maximum Likelihood Estimator for Finite Mixtures of Location–Scale Distributions When Penalty is Imposed on the Ratios of the Scale Parameters

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  • KENTARO TANAKA

Abstract

. In finite mixtures of location–scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyse the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves.

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  • Kentaro Tanaka, 2009. "Strong Consistency of the Maximum Likelihood Estimator for Finite Mixtures of Location–Scale Distributions When Penalty is Imposed on the Ratios of the Scale Parameters," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 171-184, March.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:1:p:171-184
    DOI: 10.1111/j.1467-9469.2008.00615.x
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    References listed on IDEAS

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    1. Gabriela Ciuperca & Andrea Ridolfi & Jérôme Idier, 2003. "Penalized Maximum Likelihood Estimator for Normal Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 45-59, March.
    2. Kentaro Tanaka & Akimichi Takemura, 2005. "Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 1-19, March.
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    Cited by:

    1. Mingxing He & Jiahua Chen, 2022. "Strong consistency of the MLE under two-parameter Gamma mixture models with a structural scale parameter," 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. 16(1), pages 125-154, March.
    2. Luis Angel García-Escudero & Alfonso Gordaliza & Francesca Greselin & Salvatore Ingrassia & Agustín Mayo-Iscar, 2018. "Eigenvalues and constraints in mixture modeling: geometric and computational issues," 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. 12(2), pages 203-233, June.
    3. Broda, Simon A. & Haas, Markus & Krause, Jochen & Paolella, Marc S. & Steude, Sven C., 2013. "Stable mixture GARCH models," Journal of Econometrics, Elsevier, vol. 172(2), pages 292-306.
    4. 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.
    5. Alexandrovich, Grigory, 2014. "A note on the article ‘Inference for multivariate normal mixtures’ by J. Chen and X. Tan," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 245-248.
    6. Mingxing He & Jiahua Chen, 2022. "Consistency of the MLE under a two-parameter Gamma mixture model with a structural shape parameter," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(8), pages 951-975, November.
    7. Langevin, R.;, 2024. "Consistent Estimation of Finite Mixtures: An Application to Latent Group Panel Structures," Health, Econometrics and Data Group (HEDG) Working Papers 24/16, HEDG, c/o Department of Economics, University of York.

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