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Strong consistency of the MLE under two-parameter Gamma mixture models with a structural scale parameter

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  • Mingxing He

    (Yunnan University)

  • Jiahua Chen

    (Yunnan University
    University of British Columbia)

Abstract

We study the strong consistency of the maximum likelihood estimator under a special finite mixture of two-parameter Gamma distributions. Somewhat surprisingly, the likelihood function under Gamma mixture with a set of independent and identically distributed observations is unbounded. There exist many sets of nonsensical parameter values at which the likelihood value is arbitrarily large. This leads to an inconsistent, or arguably undefined, maximum likelihood estimator. Interestingly, when the scale or shape parameter in the finite Gamma mixture model is structural, the maximum likelihood estimator of the mixing distribution is well defined and strongly consistent. Establishing the consistency when the shape parameter is structural is technically less challenging and already given in the literature. In this paper, we prove the consistency when the scale parameter is structural and provide some illustrative simulation experiments. We further include an application example of the model with a structural scale parameter to salary potential data. We conclude that the Gamma mixture distribution with a structural scale parameter provides another flexible yet relatively parsimonious model for observations with intrinsic positive values.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:advdac:v:16:y:2022:i:1:d:10.1007_s11634-021-00472-5
    DOI: 10.1007/s11634-021-00472-5
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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. Simon Lee & X. Lin, 2010. "Modeling and Evaluating Insurance Losses Via Mixtures of Erlang Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(1), pages 107-130.
    3. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504.
    4. Gordon E. Willmot & X. Sheldon Lin, 2011. "Risk modelling with the mixed Erlang distribution," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 27(1), pages 2-16, January.
    5. Xin Liu & Cristian Pasarica & Yongzhao Shao, 2003. "Testing Homogeneity in Gamma Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 227-239, March.
    6. Chen, Jiahua & Tan, Xianming, 2009. "Inference for multivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1367-1383, August.
    7. 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.
    8. Yin, Cuihong & Sheldon Lin, X. & Huang, Rongtan & Yuan, Haili, 2019. "On the consistency of penalized MLEs for Erlang mixtures," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 12-20.
    9. 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.
    10. Verbelen, Roel & Gong, Lan & Antonio, Katrien & Badescu, Andrei & Lin, Sheldon, 2015. "Fitting Mixtures Of Erlangs To Censored And Truncated Data Using The Em Algorithm," ASTIN Bulletin, Cambridge University Press, vol. 45(3), pages 729-758, September.
    11. Wong, Tony Siu Tung & Li, Wai Keung, 2014. "Test for homogeneity in gamma mixture models using likelihood ratio," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 127-137.
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