The Annals of Statistics

Current status data with competing risks: Consistency and rates of convergence of the MLE

Piet Groeneboom, Marloes H. Maathuis, and Jon A. Wellner

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We study nonparametric estimation of the sub-distribution functions for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider a simpler “naive estimator.” Both types of estimators were studied by Jewell, van der Laan and Henneman [Biometrika (2003) 90 183–197], but little was known about their large sample properties. We have started to fill this gap, by proving that the estimators are consistent and converge globally and locally at rate n1/3. We also show that this local rate of convergence is optimal in a minimax sense. The proof of the local rate of convergence of the MLE uses new methods, and relies on a rate result for the sum of the MLEs of the sub-distribution functions which holds uniformly on a fixed neighborhood of a point. Our results are used in Groeneboom, Maathuis and Wellner [Ann. Statist. (2008) 36 1064–1089] to obtain the local limiting distributions of the estimators.

Article information

Ann. Statist., Volume 36, Number 3 (2008), 1031-1063.

First available in Project Euclid: 26 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Survival analysis current status data competing risks maximum likelihood consistency rate of convergence


Groeneboom, Piet; Maathuis, Marloes H.; Wellner, Jon A. Current status data with competing risks: Consistency and rates of convergence of the MLE. Ann. Statist. 36 (2008), no. 3, 1031--1063. doi:10.1214/009053607000000974.

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  • [1] Groeneboom, P. (1996). Lectures on inverse problems. Lectures on Probability Theory and Statistics. Ecole dEté de Probabilités de Saint Flour XXIV—1994. Lecture Notes in Math. 1648 67–164. Springer, Berlin.
  • [2] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
  • [3] Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008). Current status data with competing risks: Limiting distribution of the MLE. Ann. Statist. 36 1064–1089.
  • [4] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel.
  • [5] Hudgens, M. G., Satten, G. A. and Longini, Jr., I. M. (2001). Nonparametric maximum likelihood estimation for competing risks survival data subject to interval censoring and truncation. Biometrics 57 74–80.
  • [6] Jewell, N. P. and van der Laan, M. J. (2004). Current status data: Review, recent developments and open problems. In Advances in Survival Analysis. Handbook of Statist. 23 625–642. North-Holland, Amsterdam.
  • [7] Jewell, N. P., van der Laan, M. J. and Henneman, T. (2003). Nonparametric estimation from current status data with competing risks. Biometrika 90 183–197.
  • [8] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
  • [9] Maathuis, M. H. (2005). Reduction algorithm for the MLE for the distribution function of bivariate interval censored data. J. Comput. Graph. Statist. 14 352–362.
  • [10] Maathuis, M. H. (2006). Nonparametric estimation for current status data with competing risks. Ph.D. thesis, Univ. Washington. Available at
  • [11] Pfanzagl, J. (1988). Consistency of maximum likelihood estimators for certain nonparametric families, in particular: Mixtures. J. Statist. Plann. Inference 19 137–158.
  • [12] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales. 1, 2nd ed. Wiley, Chichester.
  • [13] Schick, A. and Yu, Q. (2000). Consistency of the GMLE with mixed case interval-censored data. Scand. J. Statist. 27 45–55.
  • [14] Van de Geer, S. A. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21 14–44.
  • [15] Van de Geer, S. A. (1996). Rates of convergence of the maximum likelihood estimator in mixture models. J. Nonparametr. Statist. 6 293–310.
  • [16] Van de Geer, S. A. (2000). Applications of Empirical Process Theory. Cambridge Univ. Press.
  • [17] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • [18] Van der Vaart, A. W. and Wellner, J. A. (2000). Preservation theorems for Glivenko–Cantelli and uniform Glivenko–Cantelli classes. In High Dimensional Probability II 115–133. Birkhäuser, Boston.
  • [19] Zeidler, E. (1985). Nonlinear Functional Analysis and its Applications III: Variational Methods and Optimization. Springer, New York.