Skip to main content
SpringerLink
Log in

Nonparametric estimation of hazard functions and their derivatives under truncation model

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

Abstract

Nonparametric kernel estimators for hazard functions and their derivatives are considered under the random left truncation model. The estimator is of the form of sum of identically distributed but dependent random variables. Exact and asymptotic expressions for the biases and variances of the estimators are derived. Mean square consistency and local asymptotic normality of the estimators are established. Adaptive local bandwidths are obtained by estimating the optimal bandwidths consistently.

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

  • Allredge, J. R. and Gates, C. E. (1985). Line transect estimators for left truncated distributions,Biometrics,41, 273–280.

    Google Scholar 

  • Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York.

    Google Scholar 

  • Chao, M. T. and Lo, S. H. (1988). Some representations of the nonparametric maximum likelihood estimators with truncated data,Ann. Statist.,16, 661–668.

    Google Scholar 

  • Csörgö, S. and Horváth, L. (1980). Random censorship from the left,Studia Sci. Math. Hungar.,15, 397–401.

    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 

  • Gu, M. G. and Lai, T. L. (1990). Functional law 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 

  • Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternative,Ann. Math. Statist.,39, 325–346.

    Google Scholar 

  • Hyde, J. (1977). Testing survival under right censoring and left truncation,Biometrika,64, 225–230.

    Google Scholar 

  • Kalbfleisch, J. D. and Lawless, J. F. (1989). Inference based on transfusion-relation AIDS,J. Amer. Statist. Assoc.,84, 360–372.

    Google Scholar 

  • Keiding, N. and Gill, R. D. (1990). Random truncation models and Markov processes,Ann. Statist.,18, 582–602.

    Google Scholar 

  • Lagakos, S. W., Barraj, L. M. and DeGruttola, V. (1988). Nonparametric analysis of truncated survival data with application to AIDS,Biometrika,75, 515–523.

    Google Scholar 

  • Lui, K. J., Lawrence, D. N., Morgan, W. M., Peterman, T. A., Haverkos, H. W. and Bergman, D. J. (1986). A model based approach for estimating the mean incubation period of transfusion associated acquired immunodeficiency syndrome,Proc. Nat. Acad. Sci. U.S.A.,88, 3051–3055.

    Google Scholar 

  • Lynden-Bell, D. (1971). A method of allowing for known observational selection in small samples applied to 3CR quasars,Monthly Notices of the Royal Astronomical Society,155, 95–118.

    Google Scholar 

  • Müller, H. G. and Wang, J. L. (1990). Locally adaptive hazard smoothing,Probab. Theory Related Fields,85, 523–538.

    Google Scholar 

  • Parzen, E. (1962). On estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.

    Google Scholar 

  • Ramlau-Hansen, H. (1983). Smoothing counting process intensities by means of kernel functions,Ann. Statist.,11, 453–466.

    Google Scholar 

  • Tanner, M. A. and Wong, W. H. (1983). The estimation of the hazard function from randomly censored data by the kernel method,Ann. Statist.,11, 989–993.

    Google Scholar 

  • Uzunogullari, U. and Wang, J. L. (1990). Nonparametric estimation of hazard functions and their derivatives under truncation model, Tech. Report, #156, University of California, Davis.

    Google Scholar 

  • Uzunogullari, U. and Wang, J. L. (1992). A comparison of hazard rate estimators for left truncated and right censored data,Biometrika,79, 297–310.

    Google Scholar 

  • Wang, M. C., Jewell, N. P. and Tsai, W. Y. (1986). Asymptotic properties of the product limit estimate under random truncation,Ann. Statist.,14, 1597–1605.

    Google Scholar 

  • Watson, G. S. and Leadbetter, M. R. (1964). Hazard Analysis II,Sankhyā Ser. A,26, 101–116.

    Google Scholar 

  • Woodroofe, M. (1985). Estimating a distribution function with truncated data,Ann. Statist.,13, 163–177.

    Google Scholar 

  • Yandell, S. B. (1983). Nonparametric inference for rates with censored survival data,Ann. Statist.,11, 1119–1135.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Industrial Engineering, Bilkent University, Ankara, Turkey

    Ülkü Gürler

  2. Division of Statistics, University of California, 95616-8705, Davis, CA, USA

    Jane-Ling Wang

Authors
  1. Ülkü Gürler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jane-Ling Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by Air Force Grant AFOSR 89-0386. Part of the work of Ülkü Gürler was done while she was a Ph.D. student at the Department of Statistics, the Wharton School of the University of Pennsylvania.

About this article

Cite this article

Gürler, Ü., Wang, JL. Nonparametric estimation of hazard functions and their derivatives under truncation model. Ann Inst Stat Math 45, 249–264 (1993). https://doi.org/10.1007/BF00775812

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00775812

Key words and phrases

Search

Navigation