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Relative efficiency and deficiency of kernel type estimators of smooth distribution functions

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Abstract

The problem is investigated whether a given kernel type estimator of a distribution function at a single point has asymptotically better performance than the empirical estimator. A representation of the relative deficiency of the empirical distribution function with respect to a kernel type estimator is established which gives a complete solution to this problem. The problem of finding optimal kernels is studied in detail.

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  • M. Falk, 1983. "Relative efficiency and deficiency of kernel type estimators of smooth distribution functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 37(2), pages 73-83, June.
    • Handle: RePEc:bla:stanee:v:37:y:1983:i:2:p:73-83
    DOI: 10.1111/j.1467-9574.1983.tb00802.x
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    Cited by:

    1. Falk, Michael & Reiss, Rolf-Dieter, 2003. "Efficient estimators and LAN in canonical bivariate POT models," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 190-207, January.
    2. Bouzebda, Salim & Slaoui, Yousri, 2022. "Nonparametric recursive method for moment generating function kernel-type estimators," Statistics & Probability Letters, Elsevier, vol. 184(C).
    3. Mammitzsch Volker, 2007. "Optimal kernels," Statistics & Risk Modeling, De Gruyter, vol. 25(2/2007), pages 1-20, April.
    4. Lemdani, Mohamed & Ould-Saïd, Elias, 2003. "-deficiency of the Kaplan-Meier estimator," Statistics & Probability Letters, Elsevier, vol. 63(2), pages 145-155, June.
    5. Lloyd, Chris J. & Yong, Zhou, 1999. "Kernel estimators of the ROC curve are better than empirical," Statistics & Probability Letters, Elsevier, vol. 44(3), pages 221-228, September.
    6. Alevizos, Filippos & Bagkavos, Dimitrios & Ioannides, Dimitrios, 2019. "Efficient estimation of a distribution function based on censored data," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 359-364.
    7. Alexandre Leblanc, 2012. "On estimating distribution functions using Bernstein polynomials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 919-943, October.
    8. Daniel Janas, 1993. "A smoothed bootstrap estimator for a studentized sample quantile," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(2), pages 317-329, June.
    9. Roussas, George G., 1995. "Asymptotic normality of a smooth estimate of a random field distribution function under association," Statistics & Probability Letters, Elsevier, vol. 24(1), pages 77-90, July.
    10. Ariane Hanebeck & Bernhard Klar, 2021. "Smooth distribution function estimation for lifetime distributions using Szasz–Mirakyan operators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(6), pages 1229-1247, December.
    11. Arup Bose & Santanu Dutta, 2022. "Kernel based estimation of the distribution function for length biased data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 269-287, April.
    12. Cervellera, C. & Macciò, D., 2011. "A numerical method for minimum distance estimation problems," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 789-800, April.
    13. Shan Luo & Gengsheng Qin, 2017. "New non-parametric inferences for low-income proportions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 599-626, June.
    14. Kouros Owzar & Pranab Kumar Sen, 2003. "Copulas: concepts and novel applications," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 323-353.
    15. Ralescu, Stefan S. & Puri, Madan L., 1996. "Weak convergence of sequences of first passage processes and applications," Stochastic Processes and their Applications, Elsevier, vol. 62(2), pages 327-345, July.

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