The Annals of Statistics

Inconsistency of bootstrap: The Grenander estimator

Bodhisattva Sen, Moulinath Banerjee, and Michael Woodroofe

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Abstract

In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate n1/3. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function f on [0, ∞), is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for f(t0), where t0 ∈ (0, ∞) is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function $\mathbb{F}_{n}$ or its least concave majorant n, does not have any weak limit in probability. We provide a set of sufficient conditions for the consistency of any bootstrap method in this example and show that bootstrapping from a smoothed version of n leads to strongly consistent estimators. The m out of n bootstrap method is also shown to be consistent while generating samples from $\mathbb{F}_{n}$ and n.

Article information

Source
Ann. Statist., Volume 38, Number 4 (2010), 1953-1977.

Dates
First available in Project Euclid: 11 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1278861239

Digital Object Identifier
doi:10.1214/09-AOS777

Mathematical Reviews number (MathSciNet)
MR2676880

Zentralblatt MATH identifier
1202.62057

Subjects
Primary: 62G09: Resampling methods 62G20: Asymptotic properties
Secondary: 62G07: Density estimation

Keywords
Decreasing density empirical distribution function least concave majorant m out of n bootstrap nonparametric maximum likelihood estimate smoothed bootstrap

Citation

Sen, Bodhisattva; Banerjee, Moulinath; Woodroofe, Michael. Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 (2010), no. 4, 1953--1977. doi:10.1214/09-AOS777. https://projecteuclid.org/euclid.aos/1278861239


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References

  • Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator. Econometrica 73 1175–1204.
  • Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H. and Tukey, J. W. (1972). Robust Estimates of Location. Princeton Univ. Press, Princeton, NJ.
  • Bickel, P. and Freedman, D. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196–1217.
  • Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.
  • Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177–197. Cambridge Univ. Press, London.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31–41.
  • Grenander, U. (1956). On the theory of mortality measurement. Part II. Skand. Aktuarietidskr. 39 125–153.
  • Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. Le Cam and R. A. Olshen, eds.) 2 539–554. IMS, Hayward, CA.
  • Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388–400.
  • Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Gebiete 34 73–85.
  • Kim, J. and Pollard, D. (1990). Cube-root asymptotics. Ann. Statist. 18 191–219.
  • Kómlos, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV’s and the sample DF.I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • Kosorok, M. (2008). Bootstrapping the Grenander estimator. In Beyond Parametrics in Interdisciplinary Research: Festschrift in Honour of Professor Pranab K. Sen (N. Balakrishnan, E. Pena and M. Silvapulle, eds.) 282–292. IMS, Beachwood, OH.
  • Lee, S. M. S. and Pun, M. C. (2006). On m out of n bootstrapping for nonstandard M-estimation with nuisance parameters. J. Amer. Statist. Assoc. 101 1185–1197.
  • Léger, C. and MacGibbon, B. (2006). On the bootstrap in cube root asymptotics. Canad. J. Statist. 34 29–44.
  • Loève, M. (1963). Probability Theory. Van Nostrand, Princeton.
  • Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. Available at http://www.stat.yale.edu/~pollard/1984book/pollard1984.pdf.
  • Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhāya Ser. A 31 23–36.
  • Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
  • Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871–880.
  • Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer, New York.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Singh, K. (1981). On asymptotic accuracy of Efron’s bootstrap. Ann. Statist. 9 1187–1195.
  • van der Vaart, A. W. and Wellner, J. A. (2000). Weak Convergence and Empirical Processes. Springer, New York.
  • Wang, X. and Woodroofe, M. (2007). A Kiefer Wolfowitz comparison theorem for Wicksell’s problem. Ann. Statist. 35 1559–1575.