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Journal of Statistical Theory and Practice

, Volume 5, Issue 2, pp 221–229 | Cite as

Variable Bandwidth Kernel Density Estimation for Censored Data

  • Kagba Suaray
Article

Abstract

It has long been recognized that accurate estimation of the density function is an important problem for inference with censored data (see Gehan, 1969). This paper presents a new kernel type estimator, which smooths at observed lifetimes inversely proportional to their density according to Abramson’s square root law. It is shown that a similar reduction in bias is achieved.

AMS Subject Classification

62G07 62N02 

Key-words

Censored data Density estimation Kaplan-Meier estimator Kernel Smoothing 

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References

  1. Abramson, I., 1982. On bandwidth variation in kernel estimates - a square root law. The Annals of Statistics, 10, 1217–1223.MathSciNetCrossRefGoogle Scholar
  2. Blum, J.R., Susarla, V., 1980. Maximal deviation theory of density and failure rate function estimates based on censored data. In Multivariate analysis, V (Proc. Fifth Internat. Sympos.), Krishnaiah, P. (Editor), Univ. Pittsburgh, 213–222.Google Scholar
  3. Breslow, N., Crowley, J., 1974. A large sample study of the life table and product limit estimates under random censorship. The Annals of Statistics, 2, 437–453.MathSciNetCrossRefGoogle Scholar
  4. Claeskens, G., Hall, P., 2002. Data sharpening for hazard rate estimation. Australian and New Zealand Journal of Statistics, 44, 277–283.MathSciNetCrossRefGoogle Scholar
  5. Diehl, S., Stute, W., 1988. Kernel density and hazard function estimation in the presence of censoring. J. Multivariate Anal., 25, 299–310.MathSciNetCrossRefGoogle Scholar
  6. Földes, A., Rejtő, L., Winter, B.B., 1981. Strong consistency properties of nonparametric estimators for randomly censored data, II - Estimation of density and failure rate. Period. Math. Hungar., 12, 15–29.MathSciNetCrossRefGoogle Scholar
  7. Gehan, E., 1969. Estimating survival functions from the life table. J. Chron. Dis., 21, 629–644.CrossRefGoogle Scholar
  8. Giné, E., Sang, H., 2010. Uniform asymptotics for kernel density estimators with variable bandwidths. J. Nonpar. Stat., To appear.Google Scholar
  9. Gulati, S., Kuhn, J., Padgett, W., 2001. Comparison of some reduced-bias kernel density estimators. Math. Sci. Res. Hot-Line, 5, 29–48.MathSciNetMATHGoogle Scholar
  10. Hall, P., Hu, T.C., Marron, J.S., 1995. Improved variable window kernel estimates of probability densities. The Annals of Statistics, 23, 1–10.MathSciNetCrossRefGoogle Scholar
  11. Hall, P., Marron, J.S., 1988. Variable Window Width Kernel Estimates of Probability Densities. Probab. Th. Rel. Fields., 80, 37–49.MathSciNetCrossRefGoogle Scholar
  12. Janssen, P., Swanepoel, J., Veraverbeke, N., 2007. Modifying the kernel distribution function estimator towards reduced bias. Statistics, 41, 93–103.MathSciNetCrossRefGoogle Scholar
  13. Jones, M.C., Signorini, D.F., 1997. A comparison of higher-order bias kernel density estimators. J. Amer. Statist. Assoc., 92, 1063–1073.MathSciNetCrossRefGoogle Scholar
  14. Kaplan, E.L., Meier, P., 1958. Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc., 53, 457–481.MathSciNetCrossRefGoogle Scholar
  15. Lawless, J.F., 2003. Statistical models and methods for lifetime data, Second Edition. John Wiley, New York.MATHGoogle Scholar
  16. Lo, S.H., Mack, Y.P., Wang, J.L., 1989. Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator. Probability Theory And Related Fields, 80(3), 461–473.MathSciNetCrossRefGoogle Scholar
  17. Marron, J.S., Padgett, W.J., 1987. Asymptotically optimal bandwidth selection for kernel density estimators from randomly right-censored samples. The Annals of Statistics, 30, 1520–1535.MathSciNetCrossRefGoogle Scholar
  18. Mckay, I., 1993. A note on bias reduction in variable-kernel density estimates. The Canadian Journal of Statistics/La Revue Canadienne de Statistique, 21, 367–375.MathSciNetCrossRefGoogle Scholar
  19. Mielniczuk, J., 1986. Some asymptotic properties of kernel estimators of a density function in case of censored data. The Annals of Statistics, 14, 766–773.MathSciNetCrossRefGoogle Scholar
  20. Nielsen, J., 2003. Variable bandwidth kernel hazard estimators. J. Nonpar. Stat., 15, 355–376.MathSciNetCrossRefGoogle Scholar
  21. Parzen, E., 1962. On estimation of a probability density function and mode. Ann. Math. Stat., 33, 1065–1076.MathSciNetCrossRefGoogle Scholar
  22. Patil, P.N., Wells, M.T., Marron, J.S. 1994. Some heuristics of kernel based estimators of ratio functions. J. Nonpar. Stat., 4, 203–209.MathSciNetCrossRefGoogle Scholar
  23. Rosenblatt, M., 1956. Remarks on some nonparametric estimates of a density function. Ann. Math. Stat., 27, 832–837.MathSciNetCrossRefGoogle Scholar
  24. Scott, D.W., 1992. Multivariate Density Estimation. John Wiley, New York.CrossRefGoogle Scholar
  25. Silverman, B.W., 1986. Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.CrossRefGoogle Scholar
  26. Stute, W., 1995. The central limit theorem under random censorship. Ann. Math. Stat., 23, 422–439.MathSciNetCrossRefGoogle Scholar
  27. Terrell, G., Scott, D., 1992. Variable kernel density estimation. The Annals of Statistics, 20, 1236–1265.MathSciNetCrossRefGoogle Scholar
  28. Wand, M.P., Jones, M.C., 1994. Kernel Smoothing. Chapman and Hall, London.MATHGoogle Scholar
  29. Yandell, B., 1983. Nonparametric inference for rates with censored survival data. The Annals of Statistics, 11, 1119–1135.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsCalifornia State UniversityLong BeachUSA

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