×

Inconsistency of bootstrap: the Grenander estimator. (English) Zbl 1202.62057

Summary: We investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate \(n^{1/3}\). The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function \(f\) on \([0, \infty )\), is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for \(f(t_{0})\), where \(t_{0} \in (0, \infty )\) 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 \(\tilde F_n\), 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 \(\tilde F_n\) 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 \(\tilde F_n\).

MSC:

62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference
62G15 Nonparametric tolerance and confidence regions
62G07 Density estimation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator. Econometrica 73 1175-1204. JSTOR: · Zbl 1152.62337 · doi:10.1111/j.1468-0262.2005.00613.x
[2] 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. · Zbl 0254.62001
[3] Bickel, P. and Freedman, D. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196-1217. · Zbl 0449.62034 · doi:10.1214/aos/1176345637
[4] Breiman, L. (1968). Probability . Addison-Wesley, Reading, MA. · Zbl 0174.48801
[5] Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177-197. Cambridge Univ. Press, London.
[6] Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31-41. · Zbl 0212.21802 · doi:10.1007/BF02868560
[7] Grenander, U. (1956). On the theory of mortality measurement. Part II. Skand. Aktuarietidskr. 39 125-153. · Zbl 0077.33715
[8] 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. · Zbl 1373.62144
[9] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. JSTOR: · Zbl 04567029 · doi:10.1198/10618600152627997
[10] Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Gebiete 34 73-85. · Zbl 0354.62035 · doi:10.1007/BF00532690
[11] Kim, J. and Pollard, D. (1990). Cube-root asymptotics. Ann. Statist. 18 191-219. · Zbl 0703.62063 · doi:10.1214/aos/1176347498
[12] 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. · Zbl 0308.60029 · doi:10.1007/BF00533093
[13] 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.
[14] 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. · Zbl 1120.62310 · doi:10.1198/016214506000000014
[15] Léger, C. and MacGibbon, B. (2006). On the bootstrap in cube root asymptotics. Canad. J. Statist. 34 29-44. · Zbl 1096.62036 · doi:10.1002/cjs.5550340104
[16] Loève, M. (1963). Probability Theory . Van Nostrand, Princeton. · Zbl 0095.12201
[17] Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling . Springer, New York. · Zbl 0931.62035
[18] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, New York. Available at http://www.stat.yale.edu/ pollard/1984book/pollard1984.pdf. · Zbl 0544.60045
[19] Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhāya Ser. A 31 23-36. · Zbl 0181.45901
[20] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference . Wiley, New York. · Zbl 0645.62028
[21] Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871-880. JSTOR: · Zbl 0547.62046 · doi:10.2307/2288718
[22] Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap . Springer, New York. · Zbl 0947.62501
[23] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics . Wiley, New York. · Zbl 1170.62365
[24] Singh, K. (1981). On asymptotic accuracy of Efron’s bootstrap. Ann. Statist. 9 1187-1195. · Zbl 0494.62048 · doi:10.1214/aos/1176345636
[25] van der Vaart, A. W. and Wellner, J. A. (2000). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[26] Wang, X. and Woodroofe, M. (2007). A Kiefer Wolfowitz comparison theorem for Wicksell’s problem. Ann. Statist. 35 1559-1575. · Zbl 1209.62082 · doi:10.1214/009053606000001604
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.