Bernoulli

  • Bernoulli
  • Volume 4, Number 4 (1998), 519-543.

Efficient estimation of analytic density under random censorship

Eduard Belitser

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Abstract

The nonparametric minimax estimation of an analytic density at a given point, under random censorship, is considered. Although the problem of estimating density is known to be irregular in a certain sense, we make some connections relating this problem to the problem of estimating smooth functionals. Under condition that the censoring is not too severe, we establish the exact limiting behaviour of the local minimax risk and propose the efficient (locally asymptotically minimax) estimator - an integral of some kernel with respect to the Kaplan-Meier estimator.

Article information

Source
Bernoulli, Volume 4, Number 4 (1998), 519-543.

Dates
First available in Project Euclid: 14 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1173883819

Mathematical Reviews number (MathSciNet)
MR1679796

Zentralblatt MATH identifier
1037.62025

Keywords
asymptotic local minimax risk density estimation Kaplan-Meier estimator kernel random censorship

Citation

Belitser, Eduard. Efficient estimation of analytic density under random censorship. Bernoulli 4 (1998), no. 4, 519--543. https://projecteuclid.org/euclid.bj/1173883819


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References

  • [1] Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. New York: Springer-Verlag.
  • [2] Diehl, S. and Stute, W. (1988) Kernel density and hazard function estimation in the presence of censoring. J. Mult. Anal., 25, 299-310.
  • [3] Fedoruk, M.V. (1977) Metod Perevala. Moscow: Nauka. (In Russian.)
  • [4] Gill, R.D. (1980) Censoring and Stochastic Integrals. Vol. 124. Amsterdam: Mathematisch Centrum.
  • [5] Gill, R.D. and Levit, B.Y. (1995) Applications of the van Trees inequality: a Bayesian Cramer-Rao bound. Bernoulli, 1, 59-79.
  • [6] Golubev, G.K. and Levit, B.Y. (1996) Asymptotically efficient estimation for analytic distributions. Math. Methods Statist., 5, 357-368.
  • [7] Gradshtein, I.S. and Ryzhik, I.M. (1980) Table of Integrals, Series, and Products. New York: Academic Press.
  • [8] Hentzschel, J. (1992) Density estimation with Laguerre series and censored samples. Statistics, 23, 49-61.
  • [9] Huang, J. and Wellner, J.A. (1995) Estimation of a monotone density and monotone hazard under random censoring. Scand. J. Statist., 22, 3-33.
  • [10] Ibragimov, I.A. and Hasminskii, R.Z. (1983) Estimation of distribution density. J. Sov. Math., 25, 40-57. (Originally published in Russian in 1980.)
  • [11] Katznelson, Y. (1976) An Introduction to Harmonic Analysis, 2nd corrected edn. New York: Dover Publications.
  • [12] Kulasekera, K.B. (1995) A bound on the L1-error of a nonparametric density estimator with censored data. Statist. Probab. Lett., 23, 233-238.
  • [13] Liu, R.C. (1996) Optimal rates, minimax estimations, and K-M based procedures for estimating functionals of a distribution under censoring. Technical Report, Department of Mathematics, Cornell University.
  • [14] Lo, S.H., Mack, Y.P. and Wang, J.L. (1989) Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator. Probab. Theory Related Fields, 80, 461-473.
  • [15] Mielniczuk, J. (1986) Some asymptotic properties of kernel estimators of a density function in case of censored data. Ann. Statist., 14, 766-773.
  • [16] Nikol'skii, S.M. (1975) Approximation of Functions of Several Variables and Imbedding Theorems. Berlin: Springer-Verlag.
  • [17] Timan, A.F. (1963) Theory of Approximation of Functions of a Real Variable. Oxford: Pergamon.
  • [18] van Trees, H.L. (1968) Detection, Estimation and Modulation Theory, Part 1. New York: Wiley.
  • [19] Weits, E. (1993) The second order optimality of a smoothed Kaplan-Meier estimator. Scand. J. Statist., 20, 111-132.
  • [20] Yang, S. (1994) A central limit theorem for functionals of the Kaplan-Meier estimator. Statist. Probab. Lett., 21, 337-345.