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Global rates of convergence of the MLE for multivariate interval censoring. (English) Zbl 1336.62128

Summary: We establish global rates of convergence of the Maximum Likelihood Estimator (MLE) of a multivariate distribution function on \({\mathbb{R}}^{d}\) in the case of (one type of) “interval censored” data. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than \(n^{-1/3}(\log n)^{\gamma}\) for \(\gamma =(5d-4)/6\).

MSC:

62H12 Estimation in multivariate analysis
62N01 Censored data models
62G07 Density estimation
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information., Ann. Math. Statist. 26 641-647. · Zbl 0066.38502 · doi:10.1214/aoms/1177728423
[2] Balabdaoui, F. and Wellner, J. A. (2012). Chernoff’s density is log-concave. Technical Report No. 512, Department of Statistics, University of Washington. available as, · Zbl 1294.60100 · doi:10.3150/12-BEJ483
[3] Betensky, R. A. and Finkelstein, D. M. (1999). A non-parametric maximum likelihood estimator for bivariate interval-censored data., Statistics in Medicine 18 3089-3010.
[4] Biau, G. and Devroye, L. (2003). On the risk of estimates for block decreasing densities., J. Multivariate Anal. 86 143-165. · Zbl 1025.62015 · doi:10.1016/S0047-259X(02)00028-3
[5] Deng, D. and Fang, H.-B. (2009). On nonparametric maximum likelihood estimations of multivariate distribution function based on interval-censored data., Comm. Statist. Theory Methods 38 54-74. · Zbl 1292.62074 · doi:10.1080/03610920802155494
[6] Dunson, D. B. and Dinse, G. E. (2002). Bayesian models for multivariate current status data with informative censoring., Biometrics 58 79-88. · Zbl 1209.62031 · doi:10.1111/j.0006-341X.2002.00079.x
[7] Gao, F. (2012). Bracketing entropy of high dimensional distributions. Technical Report, Department of Mathematics, University of Idaho. “High Dimensional Probability VI”, to, appear. · Zbl 1270.41006
[8] Gentleman, R. and Vandal, A. C. (2002). Nonparametric estimation of the bivariate CDF for arbitrarily censored data., Canad. J. Statist. 30 557-571. · Zbl 1018.62022 · doi:10.2307/3316096
[9] Geskus, R. and Groeneboom, P. (1999). Asymptotically optimal estimation of smooth functionals for interval censoring, case \(2\)., Ann. Statist. 27 627-674. · Zbl 0954.62034 · doi:10.1214/aos/1018031211
[10] Groeneboom, P. (1987). Asymptotics for interval censored observations. Technical Report No. 87-18, Department of Mathematics, University of, Amsterdam.
[11] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions., Probab. Theory Related Fields 81 79-109. · doi:10.1007/BF00343738
[12] Groeneboom, P. (1996). Lectures on inverse problems. In, Lectures on probability theory and statistics (Saint-Flour, 1994) . Lecture Notes in Math. 1648 67-164. Springer, Berlin. · Zbl 0907.62042 · doi:10.1007/BFb0095675
[13] Groeneboom, P. (2012a). The bivariate current status model. Technical Report No. ??, Delft Institute of Applied Mathematics, Delft University of Technology. available as, · Zbl 1294.62062 · doi:10.1214/13-EJS824
[14] Groeneboom, P. (2012b). Local minimax lower bounds for the bivariate current status model. Technical Report No. ??, Delft Institute of Applied Mathematics, Delft University of Technology. Personal, communication. · Zbl 1253.62034
[15] Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008a). Current status data with competing risks: consistency and rates of convergence of the MLE., Ann. Statist. 36 1031-1063. · Zbl 1360.62123 · doi:10.1214/009053607000000974
[16] Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008b). Current status data with competing risks: limiting distribution of the MLE., Ann. Statist. 36 1064-1089. · Zbl 1216.62047 · doi:10.1214/009053607000000983
[17] Groeneboom, P. and Wellner, J. A. (1992)., Information bounds and nonparametric maximum likelihood estimation . DMV Seminar 19 . Birkhäuser Verlag, Basel. · Zbl 0757.62017
[18] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution., J. Comput. Graph. Statist. 10 388-400. · Zbl 04567029 · doi:10.1198/10618600152627997
[19] Jewell, N. P. (2007). Correspondences between regression models for complex binary outcomes and those for structured multivariate survival analyses. In, Advances in statistical modeling and inference . Ser. Biostat. 3 45-64. World Sci. Publ., Hackensack, NJ. · doi:10.1142/9789812708298_0003
[20] Lin, X. and Wang, L. (2011). Bayesian proportional odds models for analyzing current status data: univariate, clustered, and multivariate., Comm. Statist. Simulation Comput. 40 1171-1181. · Zbl 1227.62015 · doi:10.1080/03610918.2011.566971
[21] Maathuis, M. H. (2005). Reduction algorithm for the NPMLE for the distribution function of bivariate interval-censored data., J. Comput. Graph. Statist. 14 352-362. · doi:10.1198/106186005X48470
[22] Maathuis, M. H. (2006)., Nonparametric estimation for current status data with competing risks . ProQuest LLC, Ann Arbor, MI Thesis (Ph.D.)-University of Washington.
[23] Pavlides, M. G. (2008)., Nonparametric estimation of multivariate monotone densities . ProQuest LLC, Ann Arbor, MI Thesis (Ph.D.)-University of Washington.
[24] Pavlides, M. G. (2012). Local asymptotic minimax theory for block-decreasing densities., J. Statist. Plann. Inference 142 2322-2329. · Zbl 1244.62053 · doi:10.1016/j.jspi.2012.03.003
[25] Pavlides, M. G. and Wellner, J. A. (2012). Nonparametric estimation of multivariate scale mixtures of uniform densities., J. Multivariate Anal. 107 71-89. · Zbl 1352.62049 · doi:10.1016/j.jmva.2012.01.001
[26] 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 · doi:10.1111/1467-9469.00177
[27] Song, S. (2001). Estimation with bivariate interval-censored data. PhD thesis, University of Washington, Department of, Statistics.
[28] Sun, J. (2006)., The Statistical Analysis of Interval-censored Failure Time Data . Statistics for Biology and Health . Springer, New York. · Zbl 1127.62090
[29] van de Geer, S. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators., Ann. Statist. 21 14-44. · Zbl 0779.62033 · doi:10.1214/aos/1176349013
[30] van de Geer, S. A. (2000)., Applications of Empirical Process Theory . Cambridge Series in Statistical and Probabilistic Mathematics 6 . Cambridge University Press, Cambridge. · Zbl 0953.62049
[31] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes . Springer Series in Statistics . Springer-Verlag, New York. · Zbl 0862.60002
[32] Wang, Y.-F. (2009)., Topics on multivariate two-stage current-status data and missing covariates in survival analysis . ProQuest LLC, Ann Arbor, MI Thesis (Ph.D.)-University of California, Davis.
[33] Yu, S., Yu, Q. and Wong, G. Y. C. (2006). Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data., J. Multivariate Anal. 97 720-732. · Zbl 1333.62129 · doi:10.1016/j.jmva.2005.07.006
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