Skip to main content
SpringerLink
Log in

Kernel Density and Hazard Rate Estimation for Censored Data under α-Mixing Condition

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We derive rates of uniform strong convergence for kernel density estimators and hazard rate estimators in the presence of right censoring. It is assumed that the failure times (survival times) form a stationary α-mixing sequence. Moreover, we show that, by an appropriate choice of the bandwidth, both estimators attain the optimal strong convergence rate known from independent complete samples. The results represent an improvement over that of Cai's paper (cf. Cai (1998b, J. Multivariate Anal., 67, 23–34)).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Belyaev, Yu. K. (1991). Asymptotical properties of nonparametric point estimators based on complexly structured reliability data with right-censoring, Statistics, 22, 589–612.

    Google Scholar 

  • Breslow, N. and Crowley, J. (1974). A large sample study of the life table and product limit estimates under random censorship, Ann. Statist., 2, 437–453.

    Google Scholar 

  • Cai, Z. (1998a). Asymptotic properties of Kaplan-Meier estimator for censored dependent data, Statist. Probab. Lett., 37, 381–389.

    Google Scholar 

  • Cai, Z. (1998b). Kernel density and hazard rate estimation for censored dependent data, J. Multivariate Anal., 67, 23–34.

    Google Scholar 

  • Cai, Z. and Roussas, G. G. (1992). Uniform strong estimation under α-mixing, with rates, Statist. Probab. Lett., 15, 47–55.

    Google Scholar 

  • Diehl, S. and Stute, W. (1988). Kernel density and hazard function estimation in the presence of censoring, J. Multivariate Anal., 25, 299–310.

    Google Scholar 

  • Doukhan, P. (1994). Mixing: Properties and Examples, Lecture Notes in Statist, 85, Springer, New York.

    Google Scholar 

  • Efron, B. (1967). The two sample problem with censored data, Proc. 5th Berkeley Symp. on Math. Statist. Prob., 831–853, University of California Press, Berkeley,.

    Google Scholar 

  • Földes, A. and Rejtö, L. (1981). A LIL type result for the product limit estimator, Z. Wahrsch. Verw. Gebiete, 56, 75–84.

    Google Scholar 

  • Gill, R. (1980). Censoring and Stochastic Integrals, Mathematical Centre Tracts, Vol. 124, Mathematics Centrum, Amsterdam.

    Google Scholar 

  • Gu, M. G. and Lai, T. L. (1990). Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation, Ann. Probab., 18, 160–189.

    Google Scholar 

  • Hentzschel, J. and Liebscher, E. (1990). Density estimators for complete and censored samples, Wissenschaftliche Zeitschrift der HfV Dresden, Special Issue 57, 32–57 (in German).

    Google Scholar 

  • Kaplan, E.L. and Meier, P. (1958). Nonparametric estimation from incomplete observations, J. Amer. Statist. Assoc., 53, 457–481.

    Google Scholar 

  • Lecoutre, J. P. and Ould-Saïd, E. (1995a). Hazard rate estimation for strong-mixing and censored processes, J. Nonparametr. Statist., 5, 83–89.

    Google Scholar 

  • Lecoutre, J. P. and Ould-Saïd, E. (1995b). Convergence of the conditional Kaplan-Meier estimate under strong mixing, J. Statist. Plann. Inference, 44, 359–369.

    Google Scholar 

  • Liebscher, E. (1995). Strong convergence of sums of ϕ-mixing random variables, Math. Methods Statist., 4, 216–229.

    Google Scholar 

  • Liebscher, E. (1996). Strong convergence of sums of α-mixing random variables with applications to density estimation, Stochastic Process. Appl., 65, 69–80.

    Google Scholar 

  • Liebscher, E. (2001). Estimation of the density and the regression function under mixing conditions, Statist. Decisions, 19, 9–26.

    Google Scholar 

  • Liebscher, E. and Schäbe, H. (1997). A generalization of the Kaplan-Meier estimator to Harris-recurrent Markov chains, Statist. Papers, 38, 63–75.

    Google Scholar 

  • Mielniczuk, J. (1986). Some asymptotic properties of kernel estimators of a density function in case of censored data, Ann. Statist., 14, 766–773.

    Google Scholar 

  • Peterson, A. V. (1977). Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions, J. Amer. Statist. Assoc., 72, 854–858.

    Google Scholar 

  • Tsai, W. Y. (1986). Estimation of survival curves from dependent censorship models via a generalized self-consistent property with nonparametric Bayesian estimation application, Ann. Statist., 14, 238–249.

    Google Scholar 

  • Voelkel, J.G. and Crowley, J. (1984). Nonparametric inference for a class of semi-Markov processes with censored observations, Ann. Statist., 12, 142–160.

    Google Scholar 

  • Xiang, X. (1994). Law of the logarithm for density and hazard rate estimation for censored data, J. Multivariate Anal., 49, 278–286.

    Google Scholar 

  • Ying, Z. and Wei, L. J. (1994). The Kaplan-Meier estimate for dependent failure time observations, J. Multivariate Anal., 50, 17–29.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Technical University Ilmenau, 98684, Ilmenau/Thür, Germany

    Eckhard Liebscher

Authors
  1. Eckhard Liebscher
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Liebscher, E. Kernel Density and Hazard Rate Estimation for Censored Data under α-Mixing Condition. Annals of the Institute of Statistical Mathematics 54, 19–28 (2002). https://doi.org/10.1023/A:1016157519826

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016157519826

Search

Navigation