Skip to main content
SpringerLink
Log in

Information bounds for nonparametric estimators of L-functionals and survival functionals under censored data

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

In the present paper we derive lower asymptotic information bounds of Cramér-Rao type for estimators of nonparametric statistical functionals. The results are based on dense differentiability and dense regularity concepts which lead to weak assumptions. As explicit examples L-estimators are treated. In addition a new rapid method for the treatment of survival functionals under randomly right censored data is presented. For instance, for the famous Kaplan-Meier and Nelson-Aalen estimators, our information bound is just the lower bound obtained earlier in the literature.

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

  • Andersen PK, Borgan O, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York

    Book  MATH  Google Scholar 

  • Beutner E, Zähle H (2010) A modified functional delta method and its application to the estimation of risk functionals. J Multivar Anal 101:2452–2463

    Article  MATH  Google Scholar 

  • Boos DD (1979) A differential for L-statistics. Ann Stat 7(5):955–959

    Article  MathSciNet  MATH  Google Scholar 

  • Deheuvels P, Mason DM (1990) Bahadur-Kiefer-type processes. Ann Probab 18:669–697

    Article  MathSciNet  MATH  Google Scholar 

  • Efron B, Johnstone I (1990) Fisher’s information in terms of the hazard rate. Ann Stat 18:38–62

    Article  MathSciNet  MATH  Google Scholar 

  • Einmahl J (1996) A short and elementary proof of the main Bahadur-Kiefer theorem. Ann Probab 24:526–531

    Article  MathSciNet  MATH  Google Scholar 

  • Gill RD (1980) Censoring and stochastic integrals. Math. Centre Tracts 124. Mathematisch Centrum, Amsterdam

    Google Scholar 

  • Hájek J, Sidák Z (1967) Theory of rank tests. Academic Press, New York

    MATH  Google Scholar 

  • Janssen A (1989) Local asymptotic normality for randomly censored data with applications to rank tests. Stat Neerl 43:109–125

    Article  MATH  Google Scholar 

  • Janssen A (1994) On local odds and hazard rate models in survival analysis. Stat Probab Lett 20:355–365

    Article  MATH  Google Scholar 

  • Janssen A (1998) Zur Asymptotik nichtparametrischer Tests. Lecture Notes. Skripten zur Stochastik Nr. 29. Gesellschaft zur Förderung der Mathematischen Statistik, Münster

    Google Scholar 

  • Janssen A (2003) A nonparametric Cramér-Rao inequality for estimators of statistical functionals. Stat Probab Lett 64:347–358

    Article  MATH  Google Scholar 

  • Janssen A, Werft W (2004) A survey about the efficiency of two-sample survival tests for randomly censored data. Mitteilungen aus dem Mathematischen Seminar Giessen 254:1–47

    MathSciNet  MATH  Google Scholar 

  • Janssen A, Ostrovski V (2005) The convolution theorem of Hájek and Le Cam - revisited. arXiv:1309.4984

  • Janssen A, Wellek S (2010) Exact linear rank tests for two-sample equivalence problems with continuous data. Stat Neerl 64:482–504

    Article  MathSciNet  Google Scholar 

  • Johansen S (1978) The product limit estimator as maximum likelihood estimator. Scand J Stat 5:195–199

    MathSciNet  MATH  Google Scholar 

  • Klein JP, Moeschberger ML (1997) Survival analysis: techniques for censored and truncated data. Springer, New York

    Book  MATH  Google Scholar 

  • Le Cam L, Yang GL (2000) Asymptotics in statistics. Some basic concepts, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Lenstra AJ (2005) Cramér-Rao revisited. Bernoulli 11:263–282

    Article  MathSciNet  MATH  Google Scholar 

  • Mason DM, Shorack GR (1992) Necessary and sufficient conditions for asymptotic normality of L-statistics. Ann Probab 20:1779–1804

    Article  MathSciNet  MATH  Google Scholar 

  • Pfanzagl J (2003) Asymptotic bounds for estimators without limit distribution. Ann Inst Stat Math 55:95–110

    MathSciNet  MATH  Google Scholar 

  • Pfanzagl J, Wefelmeyer W (1982) Contributions to a general asymptotic statistical theory. Lecture notes in statistics 13. Springer, Berlin

    Book  Google Scholar 

  • Pfanzagl J, Wefelmeyer W (1985) Asymptotic expansions for general statistical models. Springer, Berlin

    Book  MATH  Google Scholar 

  • Ritov Y, Wellner JA (1988) Censoring, martingales and the Cox model. In: Pabhu NU (ed) Statistical inference from stochastic processes 80 (Contemporary Math Amer Math Soc, Providence, R.I), pp 191–219

  • Shorack GR (2000) Probability for statisticians. Springer, New York

    MATH  Google Scholar 

  • Shorack GR, Wellner JA (1986) Empirical processes with applications to statistics. Wiley, New York

    MATH  Google Scholar 

  • Stigler SM (1974) Linear functions of order statistics with smooth weight functions. Ann Stat 2:676–693

    Article  MathSciNet  MATH  Google Scholar 

  • van der Vaart AW (1988) Statistical estimation in large parameter spaces. CWI tract 44, Centrum voor Wiskunde en Informatica, Amsterdam

  • van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge

  • Wellner JA (1982) Asymptotic optimality of the product limit estimator. Ann Stat 10:595–602

    Article  MathSciNet  MATH  Google Scholar 

  • Wellner JA (2012) Private communication in April 2012

  • Witting H, Müller-Funk U (1995) Mathematische Statistik II. Teubner, Stuttgart

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut Universitätsstrasse 1, 40225, Düsseldorf, Germany

    Arnold Janssen & Andreas Knoch

Authors
  1. Arnold Janssen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Knoch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnold Janssen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Janssen, A., Knoch, A. Information bounds for nonparametric estimators of L-functionals and survival functionals under censored data. Metrika 79, 195–220 (2016). https://doi.org/10.1007/s00184-015-0551-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-015-0551-y

Keywords

Search

Navigation