# zbMATH — the first resource for mathematics

Bootstrap consistency for general semiparametric $$M$$-estimation. (English) Zbl 1200.62042
Summary: Consider $$M$$-estimation in a semiparametric model that is characterized by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. As a general purpose approach to statistical inferences, the bootstrap has found wide applications in semiparametric $$M$$-estimation and, because of its simplicity, provides an attractive alternative to the inference approach based on the asymptotic distribution theory.
The purpose of this paper is to provide theoretical justifications for the use of bootstrap as a semiparametric inferential tool. We show that, under general conditions, the bootstrap is asymptotically consistent in estimating the distribution of the $$M$$-estimate of the Euclidean parameter; that is, the bootstrap distribution asymptotically imitates the distribution of the $$M$$-estimate. We also show that the bootstrap confidence set has the asymptotically correct coverage probability. These general conclusions hold, in particular, when the nuisance parameter is not estimable at root-$$n$$ rate, and apply to a broad class of bootstrap methods with exchangeable bootstrap weights. This paper provides a first general theoretical study of the bootstrap in semiparametric models.

##### MSC:
 62G09 Nonparametric statistical resampling methods 62F40 Bootstrap, jackknife and other resampling methods 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation 62F10 Point estimation
BayesDA
Full Text:
##### References:
 [1] Barbe, P. and Bertail, P. (1995). The Weighted Bootstrap. Lecture Notes in Statistics 98 . Springer, New York. · Zbl 0826.62030 [2] Banerjee, M., Mukherjee, D. and Mishra, S. (2009). Semiparametric binary regression models under shape constraints with an application to Indian schooling data. J. Econometrics 149 101-117. · Zbl 1429.62136 [3] Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196-1217. · Zbl 0449.62034 [4] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1998). Efficient and Adaptive Estimation for Semiparametric Models . Springer, New York. · Zbl 0894.62005 [5] Chatterjee, S. and Bose, A. (2005). Generalized bootstrap for estimating equations. Ann. Statist. 33 414-436. · Zbl 1065.62073 [6] Chen, X. and Pouzo, D. (2009). Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals. J. Econometrics 152 46-60. · Zbl 1431.62111 [7] Cheng, G. (2008). Semiparametric additive isotonic regression. J. Statist. Plann. Inference 100 345-362. · Zbl 1159.62025 [8] Cheng, G. and Kosorok, M. (2008). Higher order semiparametric frequentist inference with the profile sampler. Ann. Statist. 36 1786-1818. · Zbl 1142.62030 [9] Cheng, G. and Kosorok, M. (2008). General frequentist properties of the posterior profile distribution. Ann. Statist. 36 1819-1853. · Zbl 1142.62031 [10] Delecroix, M., Hristache, M. and Patilea, V. (2006). On semiparametric M -estimation in single-index regression. J. Statist. Plann. Inference 136 730-769. · Zbl 1077.62027 [11] Dixon, J., Kosorok, M. and Lee, B. L. (2005). Functional inference in semiparametric models using the piggyback bootstrap. Ann. Inst. Statist. Math. 57 255-277. · Zbl 1082.62038 [12] Dudley, R. M. (1984). A Course on Empirical Processes. Lecture Notes in Math. 1097 2-142. Springer, Berlin. · Zbl 0554.60029 [13] Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans . SIAM, Philadelphia. · Zbl 0496.62036 [14] Efron, B. and Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statist. Sci. 1 54-75. · Zbl 0587.62082 [15] Grenander, U. (1981). Abstract Inference . Wiley, New York. · Zbl 0505.62069 [16] Gelman, A., Carlin, J., Stern, H. and Rubin, D. (2003). Bayesian Data Analysis , 2nd ed. Chapman and Hall, London. · Zbl 1279.62004 [17] Gine, E. and Zinn, J. (1990). Bootstrapping general empirical functions. Ann. Probab. 18 851-869. · Zbl 0706.62017 [18] Hall, P. (1992). The Bootstrap and Edgeworth Expansion . Springer, New York. · Zbl 0744.62026 [19] Hardle, W., Huet, S., Mammen, E. and Sperlich, S. (2004). Bootstrap inference in semiparametric generalized additive models. Econometric Theory 20 265-300. JSTOR: · Zbl 1072.62034 [20] Huang, J. (1999). Efficient estimation of the partly linear Cox model. Ann. Statist. 27 1536-1563. · Zbl 0977.62035 [21] Kosorok, M., Lee, B. L. and Fine, J. P. (2004). Robust inference for univariate proportional hazards frailty regression models. Ann. Statist. 32 1448-1491. · Zbl 1047.62090 [22] Kosorok, M. (2008). Introduction to Empirical Processes and Semiparametric Inference . Springer, New York. · Zbl 1180.62137 [23] Kosorok, M. (2008). Boostrapping the Grenander estimator. In Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen. IMS Collections 1 282-292. IMS, Beachwood, OH. [24] Lee, B. L., Kosorok, M. R. and Fine, J. P. (2005). The profile sampler. J. Amer. Statist. Assoc. 100 960-969. · Zbl 1117.62380 [25] 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 [26] Liang, H., Härdle, W. and Sommerfeld, V. (2000). Bootstrap approximations in a partially linear regression model. J. Statist. Plann. Inference 91 413-426. · Zbl 0965.62034 [27] Lo, A. Y. (1993). A Bayesian bootstrap for censored data. Ann. Statist. 21 100-123. · Zbl 0787.62048 [28] Ma, S. and Kosorok, M. (2005). Robust semiparametric M -estimation and the weighted bootstrap. J. Multivariate Anal. 96 190-217. · Zbl 1073.62030 [29] Ma, S. and Kosorok, M. (2005). Penalized log-likelihood estimation for partly linear transformation models with current status data. Ann. Statist. 33 2256-2290. · Zbl 1086.62056 [30] Mason, D. and Newton, M. (1992). A rank statistic approach to the consistency of a general bootstrap. Ann. Statist. 20 1611-1624. · Zbl 0777.62045 [31] Murphy, S. A. and van der Vaart, A. W. (1999). Observed information in semiparametric models. Bernoulli 5 381-412. · Zbl 0954.62036 [32] Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 1461-1474. JSTOR: · Zbl 0995.62033 [33] Praestgaard, J. and Wellner, J. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probab. 21 2053-2086. · Zbl 0773.60042 [34] Rubin, D. (1981). The Bayesian bootstrap. Ann. Statist. 9 130-134. [35] Sen, B., Banerjee, M. and Woodroofe, M. B. (2010). Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 1953-1977. · Zbl 1202.62057 [36] Singh, K. (1981). On the asymptotic accuracy of Efron’s bootstrap. Ann. Statist. 9 1187-1195. · Zbl 0494.62048 [37] Strawderman, R. (2006). A regression model for dependent gap times. Int. J. Biostat. 2 Article 1, 34 pp. (electronic). · Zbl 1117.62490 [38] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics . Springer, New York. · Zbl 0862.60002 [39] van der Vaart, A. W. (1998). Asymptotic Statistics . Cambridge Univ. Press, Cambridge. · Zbl 0910.62001 [40] van de Geer, S. (2000). Empirical Processes in M-Estimation . Cambridge Univ. Press, Cambridge. · Zbl 0953.62049 [41] Wellner, J. A. and Zhan, Y. (1996). Bootstrapping Z-estimators. Technical Report 308, Univ. Washington. [42] Wellner, J. A. and Zhang, Y. (2007). Two likelihood-based semiparametric estimation methods for panel count data with covariates. Ann. Statist. 35 2106-2142. · Zbl 1126.62084 [43] Young, J. G., Jewell, N. P. and Samuels, S. J. (2008). Regression analysis of a disease onset distribution using diagnosis data. Biometrics 64 20-28. · Zbl 1274.62913 [44] Zeng, D. L. and Lin, D. Y. (2007). Maximum likelihood estimation in semiparametric models with censored data (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 69 507-564. [45] Zhang, C. M. and Yu, T. (2008). Semiparametric detection of significant activation for brain FMRI. Ann. Statist. 36 1693-1725. · Zbl 1142.62026
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.