## Current status data with competing risks: Limiting distribution of the MLE.(English)Zbl 1216.62047

Summary: We study nonparametric estimation 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.” In our paper, ibid., 1031-1063 (2008; Zbl 1360.62123) we proved that both types of estimators converge globally and locally at rate $$n^{1/3}$$. We use these results to derive the local limiting distributions of the estimators. The limiting distribution of the naive estimator is given by the slopes of the convex minorants of correlated Brownian motion processes with parabolic drifts. The limiting distribution of the MLE involves a new self-induced limiting process. Finally, we present a simulation study showing that the MLE is superior to the naive estimator in terms of mean squared error, both for small sample sizes and asymptotically.

### MSC:

 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 62N02 Estimation in survival analysis and censored data 62M99 Inference from stochastic processes 62G20 Asymptotic properties of nonparametric inference 65C60 Computational problems in statistics (MSC2010)

Zbl 1360.62123
Full Text:

### References:

 [1] Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions. The Theory and Application of Isotonic Regression . Wiley, New York. · Zbl 0246.62038 [2] Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39 1563-1572. · Zbl 0169.20602 [3] Gordon, R. D. (1941). Values of Mills’ ratio of area to bounding ordinate and of the normal probability integral for large values of the argument. Ann. Math. Statist. 12 364-366. · Zbl 0026.33201 [4] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109. [5] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion + t 4 . Ann. Statist. 29 1620-1652. · Zbl 1043.62026 [6] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. · Zbl 1043.62027 [7] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2008). The support reduction algorithm for computing nonparametric function estimates in mixture models. Scand. J. Statist. To appear. Available at · Zbl 1199.65017 [8] Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008). Current status data with competing risks: Consistency and rates of convergence of the MLE. Ann. Statist. 36 1031-1063. · Zbl 1360.62123 [9] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation . Birkhäuser, Basel. · Zbl 0757.62017 [10] Hudgens, M. G., Satten, G. A. and Longini, 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 [11] Jewell, N. P. and Kalbfleisch, J. D. (2004). Maximum likelihood estimation of ordered multinomial parameters. Biostatistics 5 291-306. · Zbl 1154.62326 [12] 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 [13] Maathuis, M. H. (2003). Nonparametric maximum likelihood estimation for bivariate censored data. Master’s thesis, Delft Univ. Technology, The Netherlands. Available at http://stat.ethz.ch/ maathuis/papers. [14] Maathuis, M. H. (2006). Nonparametric estimation for current status data with competing risks. Ph.D. dissertation, Univ. Washington. Available at http://stat.ethz.ch/ maathuis/papers. [15] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, New York. Available at http://ameliabedelia.library.yale.edu/dbases/pollard1984.pdf. · Zbl 0544.60045 [16] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics . Wiley, New York. · Zbl 1170.62365 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.