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Least-squares estimation of a convex discrete distribution

Author

Listed:
  • Durot, Cécile
  • Huet, Sylvie
  • Koladjo, François
  • Robin, Stéphane

Abstract

The least squares estimator of a discrete distribution under the constraint of convexity is introduced. Its existence and uniqueness are shown and consistency and rate of convergence are established. Moreover it is shown that it always outperforms the classical empirical estimator in terms of the Euclidean distance. Results are given both in the well- and the mis-specified cases. The performance of the estimator is checked throughout a simulation study. An algorithm, based on the support reduction algorithm, is provided. Application to the estimation of species abundance distribution is discussed.

Suggested Citation

  • Durot, Cécile & Huet, Sylvie & Koladjo, François & Robin, Stéphane, 2013. "Least-squares estimation of a convex discrete distribution," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 282-298.
    • Handle: RePEc:eee:csdana:v:67:y:2013:i:c:p:282-298
    DOI: 10.1016/j.csda.2013.04.019
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    References listed on IDEAS

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    1. Dankmar Böhning & Ronny Kuhnert, 2006. "Equivalence of Truncated Count Mixture Distributions and Mixtures of Truncated Count Distributions," Biometrics, The International Biometric Society, vol. 62(4), pages 1207-1215, December.
    2. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    3. Piet Groeneboom & Geurt Jongbloed & Jon A. Wellner, 2008. "The Support Reduction Algorithm for Computing Non‐Parametric Function Estimates in Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 385-399, September.
    4. Lanumteang, K. & Böhning, D., 2011. "An extension of Chao's estimator of population size based on the first three capture frequency counts," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2302-2311, July.
    5. Dümbgen, Lutz & Rufibach, Kaspar, 2011. "logcondens: Computations Related to Univariate Log-Concave Density Estimation," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 39(i06).
    6. Walther G., 2002. "Detecting the Presence of Mixing with Multiscale Maximum Likelihood," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 508-513, June.
    7. Wang, Ji-Ping, 2011. "SPECIES: An R Package for Species Richness Estimation," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 40(i09).
    8. Fadoua Balabdaoui & Hanna Jankowski & Kaspar Rufibach & Marios Pavlides, 2013. "Asymptotics of the discrete log-concave maximum likelihood estimator and related applications," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 769-790, September.
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    Cited by:

    1. Yogesh Mani Tripathi & Amulya Kumar Mahto & Sanku Dey, 2017. "Efficient Estimation of the PDF and the CDF of a Generalized Logistic Distribution," Annals of Data Science, Springer, vol. 4(1), pages 63-81, March.
    2. Balabdaoui, Fadoua & Durot, Cécile & Koladjo, Babagnidé François, 2018. "Testing convexity of a discrete distribution," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 8-13.
    3. Balabdaoui, Fadoua & Kulagina, Yulia, 2020. "Completely monotone distributions: Mixing, approximation and estimation of number of species," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    4. Giguelay, J. & Huet, S., 2018. "Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 96-115.
    5. Sanku Dey & Tanmay Kayal & Yogesh Mani Tripathi, 2018. "Evaluation and Comparison of Estimators in the Gompertz Distribution," Annals of Data Science, Springer, vol. 5(2), pages 235-258, June.
    6. Balabdaoui, Fadoua & Durot, Cécile, 2015. "Marshall lemma in discrete convex estimation," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 143-148.

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