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Finite mixtures of unimodal beta and gamma densities and the $$k$$ -bumps algorithm

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  • Luca Bagnato
  • Antonio Punzo

Abstract

This paper addresses the problem of estimating a density, with either a compact support or a support bounded at only one end, exploiting a general and natural form of a finite mixture of distributions. Due to the importance of the concept of multimodality in the mixture framework, unimodal beta and gamma densities are used as mixture components, leading to a flexible modeling approach. Accordingly, a mode-based parameterization of the components is provided. A partitional clustering method, named $$k$$ -bumps, is also proposed; it is used as an ad hoc initialization strategy in the EM algorithm to obtain the maximum likelihood estimation of the mixture parameters. The performance of the $$k$$ -bumps algorithm as an initialization tool, in comparison to other common initialization strategies, is evaluated through some simulation experiments. Finally, two real applications are presented. Copyright Springer-Verlag Berlin Heidelberg 2013

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  • Luca Bagnato & Antonio Punzo, 2013. "Finite mixtures of unimodal beta and gamma densities and the $$k$$ -bumps algorithm," Computational Statistics, Springer, vol. 28(4), pages 1571-1597, August.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:4:p:1571-1597
    DOI: 10.1007/s00180-012-0367-4
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    1. Calabrese, Raffaella & Zenga, Michele, 2010. "Bank loan recovery rates: Measuring and nonparametric density estimation," Journal of Banking & Finance, Elsevier, vol. 34(5), pages 903-911, May.
    2. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 471-480, September.
    3. Bruche, Max & González-Aguado, Carlos, 2010. "Recovery rates, default probabilities, and the credit cycle," Journal of Banking & Finance, Elsevier, vol. 34(4), pages 754-764, April.
    4. Peter Congdon, 1993. "Statistical Graduation in Local Demographic Analysis and Projection," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 156(2), pages 237-270, March.
    5. Leisch, Friedrich, 2004. "FlexMix: A General Framework for Finite Mixture Models and Latent Class Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 11(i08).
    6. 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.
    7. Sonia Petrone, 1999. "Random Bernstein Polynomials," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 373-393, September.
    8. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    9. Schilling M.F. & Watkins A.E. & Watkins W., 2002. "Is Human Height Bimodal?," The American Statistician, American Statistical Association, vol. 56, pages 223-229, August.
    10. Biernacki, Christophe & Celeux, Gilles & Govaert, Gerard, 2003. "Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 561-575, January.
    11. Celeux, Gilles & Govaert, Gerard, 1992. "A classification EM algorithm for clustering and two stochastic versions," Computational Statistics & Data Analysis, Elsevier, vol. 14(3), pages 315-332, October.
    12. J. Wessels, 1964. "Multimodality in a family of probability densities, with application to a linear mixture of two normal densities," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 18(3), pages 267-282, September.
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