Electronic Journal of Statistics

The nonparametric bootstrap for the current status model

Piet Groeneboom and Kim Hendrickx

Full-text: Open access

Abstract

It has been proved that direct bootstrapping of the nonparametric maximum likelihood estimator (MLE) of the distribution function in the current status model leads to inconsistent confidence intervals. We show that bootstrapping of functionals of the MLE can however be used to produce valid intervals. To this end, we prove that the bootstrapped MLE converges at the right rate in the $L_{p}$-distance. We also discuss applications of this result to the current status regression model.

Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 3446-3484.

Dates
Received: January 2017
First available in Project Euclid: 6 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1507255611

Digital Object Identifier
doi:10.1214/17-EJS1345

Mathematical Reviews number (MathSciNet)
MR3709860

Zentralblatt MATH identifier
1373.62178

Subjects
Primary: 62G09: Resampling methods 62N01: Censored data models

Keywords
Bootstrap current status MLE smooth functionals

Rights
Creative Commons Attribution 4.0 International License.

Citation

Groeneboom, Piet; Hendrickx, Kim. The nonparametric bootstrap for the current status model. Electron. J. Statist. 11 (2017), no. 2, 3446--3484. doi:10.1214/17-EJS1345. https://projecteuclid.org/euclid.ejs/1507255611


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References

  • [1] Abrevaya, J. (1999). Rank regression for current-status data: asymptotic normality., Statist. Probab. Lett. 43 275–287.
  • [2] Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator., Econometrica 73 1175–1204.
  • [3] Balabdaoui, F., Groeneboom, P. and Hendrickx, K. (2017). Score estimation in the monotone single index model. working, paper.
  • [4] Banerjee, M. and Wellner, J. A. (2005). Confidence intervals for current status data., Scand. J. Statist. 32 405–424.
  • [5] Cheng, G., Huang, J. Z. et al. (2010). Bootstrap consistency for general semiparametric M-estimation., Ann. Statist. 38 2884–2915.
  • [6] Chernoff, H. (1964). Estimation of the mode., Ann. Inst. Statist. Math. 16 31–41.
  • [7] Durot, C., Groeneboom, P. and Lopuhaä, H. P. (2013). Testing equality of functions under monotonicity constraints., J. Nonparametr. Stat. 25 939–970.
  • [8] Durot, C. and Reboul, L. (2010). Goodness-of-Fit Test for Monotone Functions., Scandinavian Journal of Statistics 37 422–441.
  • [9] Efron, B. (1979). Bootstrap methods: another look at the jackknife., The Annals of Statistics 7 1-26.
  • [10] Grenander, U. (1956). On the theory of mortality measurement. II., Skand. Aktuarietidskr. 39 125–153 (1957).
  • [11] Groeneboom, P. (2012). Likelihood Ratio Type Two-Sample Tests for Current Status Data., Scandinavian Journal of Statistics 39 645–662.
  • [12] Groeneboom, P. (2014). Maximum smoothed likelihood estimators for the interval censoring model., Ann. Statist. 42 2092–2137.
  • [13] Groeneboom, P. (2015). Rcpp scripts., https://github.com/pietg/book/tree/master/Rcpp_scripts.
  • [14] Groeneboom, P. and Hendrickx, K. Confidence Intervals for the Current Status Model., Scand. J. Statist. 10.1111/sjos.12294.
  • [15] Groeneboom, P. and Hendrickx, K. (2017). Current status linear regression. Accepted for publication in Ann. Statist., available at, https://arxiv.org/abs/1601.00202.
  • [16] Groeneboom, P., Jongbloed, G. and Witte, B. I. (2010). Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model., Ann. Statist. 38 352–387.
  • [17] Groeneboom, P. and Jongbloed, G. (2014)., Nonparametric Estimation under Shape Constraints. Cambridge Univ. Press, Cambridge.
  • [18] Groeneboom, P. and Jongbloed, G. (2015). Nonparametric confidence intervals for monotone functions., Ann. Statist. 43 2019–2054.
  • [19] Groeneboom, P. and Wellner, J. A. (1992)., Information bounds and nonparametric maximum likelihood estimation. DMV Seminar 19. Birkhäuser Verlag, Basel.
  • [20] Hall, P. (1990). Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems., J. Multivariate Anal. 32 177–203.
  • [21] Hall, P. (1992a)., The bootstrap and Edgeworth expansion. Springer Series in Statistics. Springer.
  • [22] Hall, P. (1992b). Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density., Ann. Statist. 20 675–694.
  • [23] Han, A. K. (1987). Non-parametric analysis of a generalized regression model: the maximum rank correlation estimator., Journal of Econometrics 35 303–316.
  • [24] Keiding, N., Begtrup, K., Scheike, T. H. and Hasibeder, G. (1996). Estimation from Current Status Data in Continuous Time., Lifetime Data Anal. 2 119–129.
  • [25] Kim, J. K. and Pollard, D. (1990). Cube root asymptotics., Ann. Statist. 18 191–219.
  • [26] Kosorok, M. R. (2008). Bootstrapping the Grenander estimator. In, Beyond parametrics in interdisciplinary research: Festschrift in honor of Professor Pranab K. Sen. Inst. Math. Stat. Collect. 1 282–292. Inst. Math. Statist., Beachwood, OH.
  • [27] Manski, C. F. (1975). Maximum score estimation of the stochastic utility model of choice., Journal of econometrics 3 205–228.
  • [28] Patra, R. K., Seijo, E. and Sen, B. (2011). A consistent bootstrap procedure for the maximum score estimator., arXiv preprint arXiv:1105.1976.
  • [29] Rousseeuw, P. J. (1984). Least median of squares regression., Journal of the American statistical association 79 871–880.
  • [30] Schuster, E. F. (1985). Incorporating support constraints into nonparametric estimators of densities., Comm. Statist. A—Theory Methods 14 1123–1136.
  • [31] Sen, B., Banerjee, M. and Woodroofe, M. B. (2010). Inconsistency of bootstrap: the Grenander estimator., Ann. Statist. 38 1953–1977.
  • [32] Sen, B. and Xu, G. (2015). Model based bootstrap methods for interval censored data., Comput. Statist. Data Anal. 81 121–129.
  • [33] Subbotin, V. (2007). Asymptotic and bootstrap properties of rank regressions. Available at SSRN:, https://ssrn.com/abstract=1028548.
  • [34] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak convergence and empirical processes. Springer Series in Statistics. Springer-Verlag, New York. With applications to statistics.