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Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds

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  • Fadoua Balabdaoui
  • Jon A. Wellner

Abstract

The classes of monotone or convex (and necessarily monotone) densities on can be viewed as special cases of the classes of k‐monotone densities on . These classes bridge the gap between the classes of monotone (1‐monotone) and convex decreasing (2‐monotone) densities for which asymptotic results are known, and the class of completely monotone (∞‐monotone) densities on . In this paper we consider non‐parametric maximum likelihood and least squares estimators of a k‐monotone density g0. We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k−1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives , at a fixed point x0 under the assumption that .

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  • Fadoua Balabdaoui & Jon A. Wellner, 2010. "Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 45-70, February.
    • Handle: RePEc:bla:stanee:v:64:y:2010:i:1:p:45-70
    DOI: 10.1111/j.1467-9574.2009.00438.x
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    References listed on IDEAS

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    1. Dragi Anevski, 2003. "Estimating the Derivative of a Convex Density," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(2), pages 245-257, May.
    2. 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.
    3. Jongbloed, Geurt, 2000. "Minimax lower bounds and moduli of continuity," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 279-284, November.
    4. Gneiting, Tilmann, 1999. "Radial Positive Definite Functions Generated by Euclid's Hat," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 88-119, April.
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    Cited by:

    1. 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.
    2. Fadoua Balabdaoui, 2014. "Global convergence of the log-concave MLE when the true distribution is geometric," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(1), pages 21-59, March.
    3. Chee, Chew-Seng & Wang, Yong, 2016. "Nonparametric estimation of species richness using discrete k-monotone distributions," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 107-118.
    4. Balabdaoui, Fadoua & Kulagina, Yulia, 2020. "Completely monotone distributions: Mixing, approximation and estimation of number of species," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    5. Chee, Chew-Seng & Wang, Yong, 2014. "Least squares estimation of a k-monotone density function," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 209-216.
    6. Chee, Chew-Seng & Wang, Yong, 2013. "Minimum quadratic distance density estimation using nonparametric mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 1-16.

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