Some inequalities for the kernel density estimator under random censorshipFootnote^∗

^∗Research Supported by the National Natural Science Foundation of China.

References

• Birgé , L. 1987 . Estimating a density under order restrictions: Nonasympototic minimax risk . Ann. Statis. , 15 ( 3 ) : 995 – 1012 .
   Web of Science ®Google Scholar
• Blum , J. R. and Susaria , V. 1980 . “ Maximal Deviation Theory of Density and Failure Rate Function Estimates Based on Censored Data ” . Edited by: Krishnaich , P. R. 213 – 222 . North-Holland : Amsterdam . Multivariate Analysis V
   Google Scholar
• Chang , M. N. and Rao , P. V. 1989 . Berry-Esseen bound for the Kaplan- Meier estimator . Commun. Statist.-Theory. Meth. , 18 ( 12 ) : 4647 – 4664 .
   Web of Science ®Google Scholar
• Csörgö , M. and Horvíth , L. 1991 . Central limit theorems for L p distances of kernel estimators of densities under random censorship . Ann. Statist. , 19 ( 4 ) : 1813 – 1837 .
   Web of Science ®Google Scholar
• Földes , A. , Rejtö , L. and Winter , B. B. 1981 . Strong consistency properties of nonparametric estimation for randomly censored data, II: Estimation of density and failure rate . Periodica Math. Hung , 12 ( 4 ) : 15 – 29 .
   Google Scholar
• Ghorai , J. K. and Pattanaik , L. M. 1990 . L1-consistency of the kernel density estimators based on randomly right censored data . Commun. Statist.-Theory Meth. , 19 ( 8 ) : 2853 – 2870 .
   Web of Science ®Google Scholar
• Gijbels and Iréne and Wang , J. L. 1993 . Strong representation of the survival function estimator for truncated and censored data with application . J. Multi. Anal. , 47 ( 2 ) : 210 – 229 .
   Web of Science ®Google Scholar
• Kaplan , E. L. and Meier , P. 1958 . Nonparametric estimation from incomplete observations . J. Amer. Statist. Assoc. , 53 ( 2 ) : 457 – 481 .
   Web of Science ®Google Scholar
• Koul , H. , Susarla , V. and Van Ryzin , J. 1981 . Regression analysis with randomly right censored data . Ann. Statist. , 9 ( 6 ) : 1276 – 1288 .
   Web of Science ®Google Scholar
• Kulasekera , K. B. 1995 . A bound on the L1 ‐error of a nonparametric density estimator with censored data . Statistics and Probability Letters , 23 ( 6 ) : 233 – 238 .
   Web of Science ®Google Scholar
• Major , P. and Rejtö , L. 1988 . Strong embedding of the estimator of the distribution function under random censorship . Ann. Statist. , 16 ( 3 ) : 1113 – 1132 .
   Web of Science ®Google Scholar
• Marron , J. S. and Padgett , W. J. 1987 . Asymptotically optimal bandwidth selection for kernel density estimators from randomly right-censored samples . Ann. Statist. , 15 ( 3 ) : 1520 – 1535 .
   Web of Science ®Google Scholar
• Mielniczuk , J. 1986 . Some asymptotic properties of kernel estimation of a density function in case of censored data . Ann. Statist. , 14 ( 3 ) : 766 – 773 .
   Web of Science ®Google Scholar
• Parzen , E. 1962 . On estimation of a probability density function and mode . Ann. Math. Statist. , 33 ( 3 ) : 1065 – 1076 .
   Google Scholar
• Prakasa Rao , B. L. S. 1987 . “ Asymptotic Theory of Statistical Inference ” . New York : John Wiley & Sons, Inc .
   Google Scholar
• Prakasa Rao , B. L. S. 1983 . “ Nonparametric Functional estimation ” . New York : Academic Press .
   Google Scholar
• Wang , Q. H. 1994 . Some Asympototic Behaviours of the Kernel Estimator of Density Function Based on Censored Data . Chinese Journal of Applied Probability and Statistics , 10 : 51 – 61 .
   Google Scholar
• Wang , Q. H. 1996 . Consistent estimators in random censorship semiparametric regression models . Science In China , 39 : 163 – 171 .
   Google Scholar
• Wang , Q. H. 1997 . The smoothed bootstrap approximation for the kernel estimator of probability density under random censorship . Acta Math. Appl. Sinica , 20 ( 3 ) : 367 – 377 .
   Google Scholar