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Estimation and model selection of heterogeneous mixture distributions: an ECME algorithm-based approach

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  • Marco Bee

    (University of Trento, Department of Economics and Management)

Abstract

Heterogeneous mixtures and spliced distributions are commonly used models for fitting skewed data, especially in loss distribution analysis. While they guarantee additional flexibility, their estimation is typically rather difficult. In this paper we develop an Expectation Conditional Maximization Either (ECME) algorithm that allows one to estimate the parameters in a completely unsupervised manner. We also propose a likelihood-ratio test for selecting between a pure lognormal and a lognormal-Pareto distribution. With respect to a recently developed profile-likelihood method, the approach has similar statistical efficiency, but is considerably faster, thus overcoming the strong computational limitations of the profile-likelihood technique. Simulation experiments and an application to vehicle insurance illustrate the effectiveness of the estimation and testing procedures. All the codes for simulating and estimating the lognormal-Pareto mixture using the methods developed in this paper are available in the LNPar R package.

Suggested Citation

  • Marco Bee, 2026. "Estimation and model selection of heterogeneous mixture distributions: an ECME algorithm-based approach," Computational Statistics, Springer, vol. 41(2), pages 1-20, February.
    • Handle: RePEc:spr:compst:v:41:y:2026:i:2:d:10.1007_s00180-026-01723-9
    DOI: 10.1007/s00180-026-01723-9
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    References listed on IDEAS

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    1. Majda Benzidia & Michel Lubrano, 2020. "A Bayesian look at American academic wages: From wage dispersion to wage compression," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 18(2), pages 213-238, June.
    2. David Scollnik, 2007. "On composite lognormal-Pareto models," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2007(1), pages 20-33.
    3. Y. Malevergne & V. Pisarenko & D. Sornette, 2009. "Gibrat’s law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal," Swiss Finance Institute Research Paper Series 09-40, Swiss Finance Institute.
    4. Hernán D. Rozenfeld & Diego Rybski & Xavier Gabaix & Hernán A. Makse, 2011. "The Area and Population of Cities: New Insights from a Different Perspective on Cities," American Economic Review, American Economic Association, vol. 101(5), pages 2205-2225, August.
    5. Bettina Grün & Tatjana Miljkovic, 2019. "Extending composite loss models using a general framework of advanced computational tools," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(8), pages 642-660, September.
    6. Guojun Gan & Emiliano A. Valdez, 2018. "Fat-Tailed Regression Modeling with Spliced Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 22(4), pages 554-573, October.
    7. Miljkovic, Tatjana & Grün, Bettina, 2016. "Modeling loss data using mixtures of distributions," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 387-396.
    8. Marco Bee & Massimo Riccaboni & Stefano Schiavo, 2011. "Pareto versus lognormal: a maximum entropy test," Department of Economics Working Papers 1102, Department of Economics, University of Trento, Italia.
    9. Jun Ma & Hugo Jales & Zhengfei Yu, 2020. "Minimum Contrast Empirical Likelihood Inference of Discontinuity in Density," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(4), pages 934-950, October.
    10. Reynkens, Tom & Verbelen, Roel & Beirlant, Jan & Antonio, Katrien, 2017. "Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 65-77.
    11. Abu Bakar, S.A. & Hamzah, N.A. & Maghsoudi, M. & Nadarajah, S., 2015. "Modeling loss data using composite models," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 146-154.
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