Generalized bootstrap for estimating equations

Generalized bootstrap for estimating equations

Snigdhansu Chatterjee and Arup Bose

Source: Ann. Statist. Volume 33, Number 1 (2005), 414-436.

Abstract

We introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. Some special cases of this generalized bootstrap are the classical bootstrap of Efron, the delete-d jackknife and variations of the Bayesian bootstrap. The use of the proposed technique is discussed in some examples. Distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained.

Primary Subjects: 62G09, 62E20

Secondary Subjects: 62G05, 62F12, 62F40, 62M99

Keywords: Estimating equations; resampling; generalized bootstrap; jackknife; Bayesian bootstrap; wild bootstrap; paired bootstrap; M-estimation; nonlinear regression; generalized linear models; dimension asymptotics

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1112967711
Digital Object Identifier: doi:10.1214/009053604000000904
Zentralblatt MATH identifier: 02182568
Mathematical Reviews number (MathSciNet): MR2157808

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References

Basawa, I. V., Godambe, V. P. and Taylor, R. L., eds. (1997). Selected Proceedings of the Symposium on Estimating Functions. IMS, Hayward, CA.

Zentralblatt MATH: 0891.00024

Bickel, P. J. and Freedman, D. A. (1983). Bootstrapping regression models with many parameters. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 28--48. Wadsworth, Belmont, CA.

Mathematical Reviews (MathSciNet): MR689736

Zentralblatt MATH: 0529.62057

Borovskikh, Yu. V. and Korolyuk, V. S. (1997). Martingale Approximation. VSP, Utrecht.

Mathematical Reviews (MathSciNet): MR1640099

Zentralblatt MATH: 0912.60063

Bose, A. and Chatterjee, S. (2002). Comparison of bootstrap and jackknife variance estimators in linear regression: Second order results. Statist. Sinica 12 575--598.

Mathematical Reviews (MathSciNet): MR1902726

Zentralblatt MATH: 0998.62012

Bose, A. and Kushary, D. (1996). Jackknife and weighted jackknife estimation of the variance of $M$-estimators in linear regression. Technical Report 96-12, Dept. Statistics, Purdue Univ.

Chatterjee, S. (1999). Generalised bootstrap techniques. Ph.D. dissertation, Indian Statistical Institute, Calcutta.

Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1--26.

Mathematical Reviews (MathSciNet): MR515681

JSTOR: links.jstor.org

Ferguson, T. S. (1996). A Course in Large Sample Theory. Chapman and Hall, London.

Mathematical Reviews (MathSciNet): MR1699953

Zentralblatt MATH: 0871.62002

Freedman, D. A. and Peters, S. C. (1984). Bootstrapping a regression equation: Some empirical results. J. Amer. Statist. Assoc. 79 97--106.

Mathematical Reviews (MathSciNet): MR742858

Godambe, V. P., ed. (1991). Estimating Functions. Clarendon, Oxford.

Mathematical Reviews (MathSciNet): MR1163992

Hu, F. (2001). Efficiency and robustness of a resampling $M$-estimator in the linear model. J. Multivariate Anal. 78 252--271.

Mathematical Reviews (MathSciNet): MR1859758

Digital Object Identifier: doi:10.1006/jmva.2000.1951

Hu, F. and Kalbfleisch, J. D. (2000). The estimating function bootstrap (with discussion). Canad. J. Statist. 28 449--499.

Mathematical Reviews (MathSciNet): MR1793106

Huet, S., Bouvier, A., Gruet, M. and Jolivet, E. (1996). Statistical Tools for Nonlinear Regression. Springer, New York.

Mathematical Reviews (MathSciNet): MR1416565

Zentralblatt MATH: 0867.62059

Lahiri, S. N. (1992). Bootstrapping $M$-estimators of a multiple linear regression parameter. Ann. Statist. 20 1548--1570.

Mathematical Reviews (MathSciNet): MR1186265

JSTOR: links.jstor.org

Lele, S. (1991). Resampling using estimating functions. In Estimating Functions (V. P. Godambe, ed.) 295--304. Clarendon, Oxford.

Mathematical Reviews (MathSciNet): MR1163992

Liu, R. Y. and Singh, K. (1992). Efficiency and robustness in resampling. Ann. Statist. 20 370--384.

Mathematical Reviews (MathSciNet): MR1150349

JSTOR: links.jstor.org

Lo, A. Y. (1991). Bayesian bootstrap clones and a biometry function. Sankhyā Ser. A 53 320--333.

Mathematical Reviews (MathSciNet): MR1189775

Mammen, E. (1989). Asymptotics with increasing dimension for robust regression with applications to the bootstrap. Ann. Statist. 17 382--400.

Mathematical Reviews (MathSciNet): MR981457

JSTOR: links.jstor.org

Mammen, E. (1992). When Does Bootstrap Work? Asymptotic Results and Simulations. Lecture Notes in Statist. 77. Springer, New York.

Zentralblatt MATH: 0760.62038

Mammen, E. (1993). Bootstrap and wild bootstrap for high-dimensional linear models. Ann. Statist. 21 255--285.

Mathematical Reviews (MathSciNet): MR1212176

JSTOR: links.jstor.org

Myers, R. H., Montgomery, D. C. and Vining, G. G. (2002). Generalized Linear Models. Wiley, New York.

Mathematical Reviews (MathSciNet): MR1863403

Zentralblatt MATH: 0996.62065

Newton, M. A. and Raftery, A. E. (1994). Approximate Bayesian inference with the weighted likelihood bootstrap (with discussion). J. Roy. Statist. Soc. Ser. B 56 3--48.

Mathematical Reviews (MathSciNet): MR1257793

Ortega, J. M. and Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

Mathematical Reviews (MathSciNet): MR273810

Zentralblatt MATH: 0241.65046

Præ stgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probab. 21 2053--2086.

Mathematical Reviews (MathSciNet): MR1245301

JSTOR: links.jstor.org

Rao, C. R. and Zhao, L. C. (1992). Approximation to the distribution of $M$-estimates in linear models by randomly weighted bootstrap. Sankhyā Ser. A 54 323--331.

Mathematical Reviews (MathSciNet): MR1216290

Rubin, D. B. (1981). The Bayesian bootstrap. Ann. Statist. 9 130--134.

Mathematical Reviews (MathSciNet): MR600538

JSTOR: links.jstor.org

Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.

Mathematical Reviews (MathSciNet): MR595165

Zentralblatt MATH: 0538.62002

Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). Ann. Statist. 14 1261--1350.

Mathematical Reviews (MathSciNet): MR868303

JSTOR: links.jstor.org

Zheng, Z. and Tu, D. (1988). Random weighting method in regression models. Sci. Sinica Ser. A 31 1442--1459.

Mathematical Reviews (MathSciNet): MR983865

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