The Annals of Statistics

Isotonic inverse estimators for nonparametric deconvolution

Geurt Jongbloed, Bert van Es, and Martien van Zuijlen

Full-text: Open access

Abstract

A new nonparametric estimation procedure is introduced for the distribution function in a class of deconvolution problems, where the convolution density has one discontinuity. The estimator is shown to be consistent and its cube root asymptotic distribution theory is established. Known results on the minimax risk for the estimation problem indicate the estimator to be efficient.

Article information

Source
Ann. Statist. Volume 26, Number 6 (1998), 2395-2406.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1024691476

Digital Object Identifier
doi:10.1214/aos/1024691476

Mathematical Reviews number (MathSciNet)
MR1700237

Zentralblatt MATH identifier
0927.62029

Subjects
Primary: 62G05
Secondary: 62E20: Asymptotic distribution theory

Keywords
Convex minorant cube root asymptotics isotonic estimation empirical process

Citation

van Es, Bert; Jongbloed, Geurt; van Zuijlen, Martien. Isotonic inverse estimators for nonparametric deconvolution. Ann. Statist. 26 (1998), no. 6, 2395--2406. doi:10.1214/aos/1024691476. http://projecteuclid.org/euclid.aos/1024691476.


Export citation

References

  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272.
  • Gripenberg, G., Londen, S.-O. and Staffans, O. (1990). Volterra Integral and Functional Equations. Cambridge Univ. Press.
  • Groeneboom, P. (1996). Lectures on inverse problems. Ecole d'Et´e de Probabilit´es de Saint-Flour XXIV. Lectures on Probability Theory and Statistics 67-164. Springer, Berlin.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkh¨auser, Basel.
  • Hall, P. and Diggle, P. J. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 523-531.
  • Jongbloed, G. (1995). Three statistical inverse problems. Ph.D. dissertation, Delft Univ. Technology.
  • Jongbloed, G. (1998). Exponential deconvolution: two asy mptotically equivalent estimators. Statist. Neerlandica 52, 6-17.
  • Robertson, T., Wright, F. T. and Dy kstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. van Es, A. J. (1991a). Aspects of nonparametric density estimation. CWI Tract 77. Centre for Mathematics and Computer Science, Amsterdam. van Es, A. J. (1991b). Uniform deconvolution: nonparametric maximum likelihood and inverse estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 191-198. Kluwer, Dordrecht.
  • van Es, B. and Kok, A. (1997). Simple kernel estimators for certain nonparametric deconvolution problems. Statist. Probab. Lett. To appear.
  • van Es, A. J. and van Zuijlen, M. C. A. (1996). Convex minorant estimators of distributions in nonparametric deconvolution problems. Scand. J. Statist. 23 85-104.
  • van Es, B., Jongbloed, G. and van Zuijlen, M. (1995). Nonparametric deconvolution for decreasing kernels. Report 95-77, Delft Univ. Technology.