The Annals of Statistics

Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model

Piet Groeneboom, Geurt Jongbloed, and Birgit I. Witte

Full-text: Open access


We consider the problem of estimating the distribution function, the density and the hazard rate of the (unobservable) event time in the current status model. A well studied and natural nonparametric estimator for the distribution function in this model is the nonparametric maximum likelihood estimator (MLE). We study two alternative methods for the estimation of the distribution function, assuming some smoothness of the event time distribution. The first estimator is based on a maximum smoothed likelihood approach. The second method is based on smoothing the (discrete) MLE of the distribution function. These estimators can be used to estimate the density and hazard rate of the event time distribution based on the plug-in principle.

Article information

Ann. Statist., Volume 38, Number 1 (2010), 352-387.

First available in Project Euclid: 31 December 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62N01: Censored data models
Secondary: 62G20: Asymptotic properties

Current status data maximum smoothed likelihood smoothed maximum likelihood distribution estimation density estimation hazard rate estimation asymptotic distribution


Groeneboom, Piet; Jongbloed, Geurt; Witte, Birgit I. Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model. Ann. Statist. 38 (2010), no. 1, 352--387. doi:10.1214/09-AOS721.

Export citation


  • Bickel, P. J. and Fan, J. (1996). Some problems on the estimation of unimodal densities. Statist. Sinica 6 23–45.
  • Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200–1224.
  • Dümbgen, L., Freitag-Wolf, S. and Jongbloed, G. (2006). Estimating a unimodal distribution function from interval censored data. J. Amer. Statist. Assoc. 101 1094–1106.
  • Dümbgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli 15 40–68.
  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
  • Eggermont, P. P. B. and LaRiccia, V. N. (2001). Maximum Penalized Likelihood Estimation. Springer, New York.
  • González-Manteiga, W., Cao, R. and Marron, J. S. (1996). Bootstrap selection of the smoothing parameter in nonparametric hazard rate estimation. J. Amer. Statist. Assoc. 91 1130–1140.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2002). A canonical process for estimation of convex functions: The invelope of integrated brownian motion +t4. Ann. Statist. 29 1620–1652.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Cambridge Univ. Press, New York.
  • Hazelton, M. L. (1996). Bandwidth selection for local density estimators. Scand. J. Statist. 23 221–232.
  • Huang, Y. and Zhang, C. H. (1994). Estimating a monotone density from censored observations. Ann. Statist. 22 1256–1274.
  • Jones, M. C. (1993). Simple boundary correction for kernel density estimation. Stat. Comput. 3 135–146.
  • Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete data. J. Amer. Statist. Assoc. 53 457–481.
  • Keiding, N. (1991). Age-specific incidence and prevalence: A statistical perspective. J. Roy. Statist. Soc. Ser. A 154 371–412.
  • Mammen, E. (1991). Estimating a monotone regression function. Ann. Statist. 19 724–740.
  • Marron, J. S. and Padgett, W. J. (1987). Asymptotically optimal bandwidth selection for kernel density estimators from randomly right-censored samples. Ann. Statist. 15 1520–1535.
  • Patil, P. N., Wells, M. T. and Marron, J. S. (1994). Some heuristics of kernel based estimators of ratio functions. J. Nonparametr. Stat. 4 203–209.
  • Schuster, E. F. (1985). Incorporating support constraints into nonparametric estimators of densities. Comm. Statist. Theory Methods 14 1123–1136.
  • Sheather, S. J. (1983). A data-based algorithm for choosing the window width when estimating the density at a point. Comput. Statist. Data Anal. 1 229–238.
  • Silverman, B. W. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivative. Ann. Statist. 6 177–184.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
  • van de Geer, S. A. (2000). Applications of Empirical Process Theory. Cambridge Univ. Press, New York.
  • van der Vaart, A. W. and van der Laan, M. J. (2003). Smooth estimation of a monotone density. Statistics 37 189–203.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.