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Estimating Smoothness and Optimal Bandwidth for Probability Density Functions

Author

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  • Dimitris N. Politis

    (Department of Mathematics and Halicioglu Data Science Institute, University of California, San Diego, CA 92093-0112, USA)

  • Peter F. Tarassenko

    (Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia)

  • Vyacheslav A. Vasiliev

    (Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia)

Abstract

The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One of the classes was introduced by Rosenblatt, another one is introduced in this paper. For the case of the known smoothness parameter, the rates of mean square convergence of optimal (on the bandwidth) density estimators are found. For the case of unknown smoothness parameter, the estimation procedure of the parameter is developed and almost surely convergency is proved. The convergence rates in the almost sure sense of these estimators are obtained. Adaptive estimators of densities from the given class on the basis of the constructed smoothness parameter estimators are presented. It is shown in examples how parameters of the adaptive density estimation procedures can be chosen. Non-asymptotic and asymptotic properties of these estimators are investigated. Specifically, the upper bounds for the mean square error of the adaptive density estimators for a fixed sample size are found and their strong consistency is proved. The convergence of these estimators in the almost sure sense is established. Simulation results illustrate the realization of the asymptotic behavior when the sample size grows large.

Suggested Citation

  • Dimitris N. Politis & Peter F. Tarassenko & Vyacheslav A. Vasiliev, 2022. "Estimating Smoothness and Optimal Bandwidth for Probability Density Functions," Stats, MDPI, vol. 6(1), pages 1-20, December.
    • Handle: RePEc:gam:jstats:v:6:y:2022:i:1:p:3-49:d:1016627
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    References listed on IDEAS

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    1. Marc Hallin & Lanh T. Tran, 1996. "Kernel density estimation for linear processes: asymptotic normality and bandwidth selection," ULB Institutional Repository 2013/2055, ULB -- Universite Libre de Bruxelles.
    2. Marc Hallin & Lanh Tran, 1996. "Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 429-449, September.
    3. Zudi Lu, 2001. "Asymptotic Normality of Kernel Density Estimators under Dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 447-468, September.
    4. Vyacheslav Vasiliev, 2014. "A truncated estimation method with guaranteed accuracy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 141-163, February.
    5. Claire Lacour & Pascal Massart & Vincent Rivoirard, 2017. "Estimator Selection: a New Method with Applications to Kernel Density Estimation," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 298-335, August.
    6. Politis, Dimitris N. & Romano, Joseph P., 1999. "Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 1-25, January.
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