## Electronic Journal of Statistics

### Global rates of convergence of the MLE for multivariate interval censoring

#### Abstract

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 $\gamma =(5d-4)/6$.

#### Article information

Source
Electron. J. Statist. Volume 7 (2013), 364-380.

Dates
First available in Project Euclid: 28 January 2013

http://projecteuclid.org/euclid.ejs/1359382684

Digital Object Identifier
doi:10.1214/13-EJS777

Mathematical Reviews number (MathSciNet)
MR3020425

Zentralblatt MATH identifier
06167940

Subjects
Primary: 62G07: Density estimation 62H12: Estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

#### Citation

Gao, Fuchang; Wellner, Jon A. Global rates of convergence of the MLE for multivariate interval censoring. Electron. J. Statist. 7 (2013), 364--380. doi:10.1214/13-EJS777. http://projecteuclid.org/euclid.ejs/1359382684.

#### References

• 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.
• 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, arXiv:1207.6614.
• 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.
• Biau, G. and Devroye, L. (2003). On the risk of estimates for block decreasing densities., J. Multivariate Anal. 86 143–165.
• 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.
• Dunson, D. B. and Dinse, G. E. (2002). Bayesian models for multivariate current status data with informative censoring., Biometrics 58 79–88.
• Gao, F. (2012). Bracketing entropy of high dimensional distributions. Technical Report, Department of Mathematics, University of Idaho. “High Dimensional Probability VI”, to, appear.
• Gentleman, R. and Vandal, A. C. (2002). Nonparametric estimation of the bivariate CDF for arbitrarily censored data., Canad. J. Statist. 30 557–571.
• Geskus, R. and Groeneboom, P. (1999). Asymptotically optimal estimation of smooth functionals for interval censoring, case $2$., Ann. Statist. 27 627–674.
• Groeneboom, P. (1987). Asymptotics for interval censored observations. Technical Report No. 87-18, Department of Mathematics, University of, Amsterdam.
• Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions., Probab. Theory Related Fields 81 79–109.
• 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.
• Groeneboom, P. (2012a). The bivariate current status model. Technical Report No. ??, Delft Institute of Applied Mathematics, Delft University of Technology. available as, arXiv:1209.0542.
• 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.
• 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.
• 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.
• Groeneboom, P. and Wellner, J. A. (1992)., Information bounds and nonparametric maximum likelihood estimation. DMV Seminar 19. Birkhäuser Verlag, Basel.
• Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution., J. Comput. Graph. Statist. 10 388–400.
• 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.
• 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.
• 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.
• Maathuis, M. H. (2006)., Nonparametric estimation for current status data with competing risks. ProQuest LLC, Ann Arbor, MI Thesis (Ph.D.)–University of Washington.
• Pavlides, M. G. (2008)., Nonparametric estimation of multivariate monotone densities. ProQuest LLC, Ann Arbor, MI Thesis (Ph.D.)–University of Washington.
• Pavlides, M. G. (2012). Local asymptotic minimax theory for block-decreasing densities., J. Statist. Plann. Inference 142 2322–2329.
• Pavlides, M. G. and Wellner, J. A. (2012). Nonparametric estimation of multivariate scale mixtures of uniform densities., J. Multivariate Anal. 107 71–89.
• Schick, A. and Yu, Q. (2000). Consistency of the GMLE with mixed case interval-censored data., Scand. J. Statist. 27 45–55.
• Song, S. (2001). Estimation with bivariate interval–censored data. PhD thesis, University of Washington, Department of, Statistics.
• Sun, J. (2006)., The Statistical Analysis of Interval-censored Failure Time Data. Statistics for Biology and Health. Springer, New York.
• van de Geer, S. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators., Ann. Statist. 21 14–44.
• van de Geer, S. A. (2000)., Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge University Press, Cambridge.
• van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer-Verlag, New York.
• 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.
• 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.