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Journal of Probability and Statistics
Volume 2011, Article ID 937574, 16 pages
Research Article

Gamma Kernel Estimators for Density and Hazard Rate of Right-Censored Data

1Département de Mathématiques, Université de Sherbrooke, Québec, QC, Canada JIK 2RI
2Institut de Statistique, Biostatistique et Sciences Actuarielles, Université Catholique de Louvain, 1348 Louvain-La-Neuve, Belgium
3Département de Mathémathique et d'Informatique, Université du Québec à Trois Rivières, Trois Rivières, QC, Canada G9A 5H7

Received 15 September 2010; Revised 29 March 2011; Accepted 18 April 2011

Academic Editor: Michael Lavine

Copyright © 2011 T. Bouezmarni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The nonparametric estimation for the density and hazard rate functions for right-censored data using the kernel smoothing techniques is considered. The “classical” fixed symmetric kernel type estimator of these functions performs well in the interior region, but it suffers from the problem of bias in the boundary region. Here, we propose new estimators based on the gamma kernels for the density and the hazard rate functions. The estimators are free of bias and achieve the optimal rate of convergence in terms of integrated mean squared error. The mean integrated squared error, the asymptotic normality, and the law of iterated logarithm are studied. A comparison of gamma estimators with the local linear estimator for the density function and with hazard rate estimator proposed by Müller and Wang (1994), which are free from boundary bias, is investigated by simulations.