## Current status data with competing risks: consistency and rates of convergence of the MLE.(English)Zbl 1360.62123

Summary: 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 N. P. Jewell et al. [Biometrika 90, No. 1, 183–197 (2003; Zbl 1034.62034)], 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 $$n^{1/3}$$. 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 our paper [Ann. Stat. 36, No. 3, 1064–1089 (2008; Zbl 1216.62047)] to obtain the local limiting distributions of the estimators.

### MSC:

 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62N01 Censored data models

### Citations:

Zbl 1034.62034; Zbl 1216.62047
Full Text:

### References:

 [1] Groeneboom, P. (1996). Lectures on inverse problems. Lectures on Probability Theory and Statistics. Ecole d ’ Eté de Probabilités de Saint Flour XXIV-1994. Lecture Notes in Math. 1648 67-164. Springer, Berlin. · Zbl 0907.62042 [2] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. · Zbl 1043.62027 [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. · Zbl 1216.62047 [4] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation . Birkhäuser, Basel. · Zbl 0757.62017 [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. JSTOR: · Zbl 1209.62049 [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. · Zbl 1034.62034 [8] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219. · Zbl 0703.62063 [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 http://stat.ethz.ch/ maathuis/papers/. [11] Pfanzagl, J. (1988). Consistency of maximum likelihood estimators for certain nonparametric families, in particular: Mixtures. J. Statist. Plann. Inference 19 137-158. · Zbl 0656.62044 [12] Rogers, L. C. G. and Williams, D. (1994). Diffusions , Markov Processes , and Martingales . 1 , 2nd ed. Wiley, Chichester. · Zbl 0826.60002 [13] Schick, A. and Yu, Q. (2000). Consistency of the GMLE with mixed case interval-censored data. Scand. J. Statist. 27 45-55. · Zbl 0938.62109 [14] Van de Geer, S. A. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21 14-44. · Zbl 0779.62033 [15] Van de Geer, S. A. (1996). Rates of convergence of the maximum likelihood estimator in mixture models. J. Nonparametr. Statist. 6 293-310. · Zbl 0872.62039 [16] Van de Geer, S. A. (2000). Applications of Empirical Process Theory . Cambridge Univ. Press. · Zbl 0953.62049 [17] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes : With Applications to Statistics . Springer, New York. · Zbl 0862.60002 [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. · Zbl 0967.60037 [19] Zeidler, E. (1985). Nonlinear Functional Analysis and its Applications III : Variational Methods and Optimization . Springer, New York. · Zbl 0583.47051
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