The Annals of Statistics

Asymptotically optimal estimation of smooth functionals for interval censoring, case $2$

Ronald Geskus and Piet Groeneboom

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Abstract

For a version of the interval censoring model, case 2, in which the observation intervals are allowed to be arbitrarily small, we consider estimation of functionals that are differentiable along Hellinger differentiable paths. The asymptotic information lower bound for such functionals can be represented as the squared $L_{2}$-norm of the canonical gradient in the observation space. This canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types. We study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator (NPMLE) , and derive the asymptotic normality and efficiency of the NPMLE from properties of the solutions of the integral equations.

Article information

Source
Ann. Statist., Volume 27, Number 2 (1999), 627-674.

Dates
First available in Project Euclid: 5 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1018031211

Digital Object Identifier
doi:10.1214/aos/1018031211

Mathematical Reviews number (MathSciNet)
MR1714713

Zentralblatt MATH identifier
0954.62034

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 62E20: Asymptotic distribution theory 62G05: Estimation 62G20: Asymptotic properties 45A05: Linear integral equations

Keywords
Nonparametric maximum likelihood empirical processes asymptotic distributions asymptotic efficiency integral equations.

Citation

Geskus, Ronald; Groeneboom, Piet. Asymptotically optimal estimation of smooth functionals for interval censoring, case $2$. Ann. Statist. 27 (1999), no. 2, 627--674. doi:10.1214/aos/1018031211. https://projecteuclid.org/euclid.aos/1018031211


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References

  • AYER, M., BRUNK, H. D., EWING, G. M., REID, W. T. and SILVERMAN, E. 1955. An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641 647. Z.
  • BALL, K. and PAJOR, A. 1990. The entropy of convex bodies with ``few'' extreme points. In Geometry of Banach Spaces 25 32. Cambridge Univ. Press. Z.
  • BERMAN, A. and PLEMMONS, R. J. 1979. Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York. Z.
  • BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. 1993. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore.
  • BIRMAN, M. S. and SOLOMJAK, A. 1967. Piecewise-polynomial approximations of functions of the classes W. Mat. Sb. 73 295 317. p Z.
  • GESKUS, R. B. 1992. Efficient estimation of the mean for interval censoring case II. Technical Report 92-83, Delft Univ. Technology, ftp: ftp.twi.tudelft.nl pub publications tech-reports 1992. Z.
  • GESKUS, R. B. 1997. Estimation of smooth functionals with interval censored data, and something completely different. Ph.D. dissertation, Delft Univ. Technology. Z.
  • GESKUS, R. B. and GROENEBOOM, P. 1995. Asymptotically optimal estimation of smooth functionals for interval censoring, case 2 and beyond. Technical Report 95-78, Delft Univ. Technology, ftp: ftp.twi.tudelft.nl pub publications tech-reports 1995. Z.
  • GESKUS, R. B. and GROENEBOOM, P. 1996. Asymptotically optimal estimation of smooth functionals for interval censoring, part 1. Statist. Neerlandica 50 69 88. Z.
  • GESKUS, R. B. and GROENEBOOM, P. 1997. Asymptotically optimal estimation of smooth functionals for interval censoring, part 2. Statist. Neerlandica 51 201 219.
  • GROENEBOOM, P. and WELLNER, J. A. 1992. Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhauser, Boston. ¨ Z.
  • GROENEBOOM, P. 1996. Lectures on inverse problems. Lectures on Probability Theory. Ecole d'Ete de Probabilites de Saint-Flour XXIV. Springer, Berlin. ´ ´Z.
  • HUANG, J. and WELLNER, J. A. 1995. Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1. Statist. Neerlandica 49 153 163. http: www.stat. washington.edu:80 jaw jaw.research.available.html. Z.
  • JONGBLOED, G. 1998. The iterative convex minorant algorithm for nonparametric estimation. J. Comput. Graph. Statist. 7 310 321. http: www.amstat.org publications jcgs toc 98.html. Z.
  • KRESS, R. 1989. Linear Integral Equations. Springer, New York. Z.
  • POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z.
  • VAN DE GEER, S. 1996. Rates of convergence for the maximum likelihood estimator in mixture models. J. Nonparametr. Statist. 6 293 310. Z.
  • VAN DER VAART, A. W. 1991. On differentiable functionals. Ann. Statist. 19 178 204. Z.
  • VAN DER VAART, A. W. and WELLNER, J. 1996. Weak Convergence and Empirical Processes. Springer, New York. Z.
  • VAN EEDEN, C. 1956. Maximum likelihood estimation of ordered probabilities. Indag. Math. 18 444 455. Z.
  • WELLNER, J. 1995. Interval censoring case 2: alternative hypotheses. In Analysis of Censored Z. Data H. L. Koul and J. V. Deshpande, eds. 271 291. IMS, Hayward, CA. http: www.stat.washington.edu:80 jaw jaw.research.available.html.