Current status data with competing risks: Limiting distribution of the MLE
Piet Groeneboom, Marloes H. Maathuis, and Jon A. Wellner
Source: Ann. Statist. Volume 36, Number 3
(2008), 1064-1089.
Abstract
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.” Groeneboom, Maathuis and Wellner [Ann. Statist. (2008) 36 1031–1063] proved that both types of estimators converge globally and locally at rate n1/3. 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.
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Keywords: Survival analysis; current status data; competing risks; maximum likelihood; limiting distribution
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1211819556
Digital Object Identifier: doi:10.1214/009053607000000983
Mathematical Reviews number (MathSciNet): MR2418649
Zentralblatt MATH identifier: 05294965
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.
Mathematical Reviews (MathSciNet): MR326887
Zentralblatt MATH: 0246.62038
[2] Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39 1563–1572.
Mathematical Reviews (MathSciNet): MR230338
Project Euclid: euclid.aoms/1177698137
[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.
Mathematical Reviews (MathSciNet): MR5558
Digital Object Identifier: doi:10.1214/aoms/1177731721
Project Euclid: euclid.aoms/1177731721
[4] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79–109.
Mathematical Reviews (MathSciNet): MR981568
Digital Object Identifier: doi:10.1007/BF00343738
[5] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion +t4. Ann. Statist. 29 1620–1652.
[6] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
Mathematical Reviews (MathSciNet): MR1891742
Digital Object Identifier: doi:10.1214/aos/1015345958
Project Euclid: euclid.aos/1015345958
Zentralblatt MATH: 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 arxiv:math/ST/0405511.
Mathematical Reviews (MathSciNet): MR2446726
Digital Object Identifier: doi:10.1111/j.1467-9469.2007.00588.x
[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.
Mathematical Reviews (MathSciNet): MR2418648
Digital Object Identifier: doi:10.1214/009053607000000974
Project Euclid: euclid.aos/1211819555
Zentralblatt MATH: 05294964
[9] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet): MR1180321
Zentralblatt MATH: 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.
Mathematical Reviews (MathSciNet): MR1821337
Digital Object Identifier: doi:10.1111/j.0006-341X.2001.00074.x
JSTOR: links.jstor.org
[11] Jewell, N. P. and Kalbfleisch, J. D. (2004). Maximum likelihood estimation of ordered multinomial parameters. Biostatistics 5 291–306.
[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.
Mathematical Reviews (MathSciNet): MR1966559
Zentralblatt MATH: 1034.62034
Digital Object Identifier: doi:10.1093/biomet/90.1.183
[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.
Mathematical Reviews (MathSciNet): MR762984
Zentralblatt MATH: 0544.60045
[16] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet): MR838963
[17] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
Mathematical Reviews (MathSciNet): MR1385671
Zentralblatt MATH: 0862.60002
