The Impact of the Bootstrap on Statistical Algorithms and Theory
Rudolf Beran
Source: Statist. Sci. Volume 18, Issue 2 (2003), 175-184.
Abstract
Bootstrap ideas yield remarkably effective algorithms for realizing certain programs in statistics. These include the construction of (possibly simultaneous) confidences sets and tests in classical models for which exact or asymptotic distribution theory is intractable. Success of the bootstrap, in the sense of doing what is expected under a probability model for data, is not universal. Modifications to Efron's definition of the bootstrap are needed to make the idea work for modern procedures that are not classically regular.
Keywords: Confidence sets; error in coverage probability; double bootstrap; simultaneous confidence sets; local asymptotic equivariance; convolution theorem
Full-text: Open access
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References
Beran, R. (1986). Simulated power functions. Ann. Statist. 14 151--173.
Mathematical Reviews (MathSciNet):
MR829560
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aos/1176349847
Project Euclid: euclid.aos/1176349847
Beran, R. (1987). Prepivoting to reduce level error of confidence sets. Biometrika 74 457--468.
Mathematical Reviews (MathSciNet):
MR909351
Zentralblatt MATH:
0663.62045
Digital Object Identifier: doi:10.2307/2336685
JSTOR: links.jstor.org
Beran, R. (1988a). Balanced simultaneous confidence sets. J. Amer. Statist. Assoc. 83 679--686.
Mathematical Reviews (MathSciNet):
MR963795
Digital Object Identifier: doi:10.2307/2289291
JSTOR: links.jstor.org
Beran, R. (1988b). Prepivoting test statistics: A bootstrap view of asymptotic refinements. J. Amer. Statist. Assoc. 83 687--697.
Mathematical Reviews (MathSciNet):
MR963796
Digital Object Identifier: doi:10.2307/2289292
JSTOR: links.jstor.org
Beran, R. (1990). Refining bootstrap simultaneous confidence sets. J. Amer. Statist. Assoc. 85 417--426.
Mathematical Reviews (MathSciNet):
MR1141742
Digital Object Identifier: doi:10.2307/2289778
JSTOR: links.jstor.org
Beran, R. (1997). Diagnosing bootstrap success. Ann. Inst. Statist. Math. 49 1--24.
Mathematical Reviews (MathSciNet):
MR1450689
Digital Object Identifier: doi:10.1023/A:1003114420352
Beran, R. and Ducharme, G. (1991). Asymptotic Theory for Bootstrap Methods in Statistics. Les Publications CRM, Univ. Montréal.
Mathematical Reviews (MathSciNet):
MR1133624
Zentralblatt MATH:
0733.62050
Beran, R. and Dümbgen, L. (1998). Modulation of estimators and confidence sets. Ann. Statist. 26 1826--1856.
Mathematical Reviews (MathSciNet):
MR1673280
Digital Object Identifier: doi:10.1214/aos/1024691359
Project Euclid: euclid.aos/1024691359
Beran, R. and Fisher, N. I. (1998). Nonparametric comparison of mean directions or mean axes. Ann. Statist. 26 472--493.
Mathematical Reviews (MathSciNet):
MR1626051
Digital Object Identifier: doi:10.1214/aos/1028144845
Project Euclid: euclid.aos/1028144845
Beran, R. and Millar, P. W. (1986). Confidence sets for a multivariate distribution. Ann. Statist. 14 431--443.
Mathematical Reviews (MathSciNet):
MR840507
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aos/1176349931
Project Euclid: euclid.aos/1176349931
Beran, R. and Srivastava, M. S. (1985). Bootstrap tests and confidence regions for functions of a covariance matrix. Ann. Statist. 13 95--115.
Mathematical Reviews (MathSciNet):
MR773155
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aos/1176346579
Project Euclid: euclid.aos/1176346579
Berlinski, D. (2001). The Advent of the Algorithm. Harcourt, New York.
Mathematical Reviews (MathSciNet):
MR1766416
Zentralblatt MATH:
0981.00004
Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196--1217.
Mathematical Reviews (MathSciNet):
MR630103
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aos/1176345637
Project Euclid: euclid.aos/1176345637
Bickel, P. J. and Ren, J.-J. (2001). The bootstrap in hypothesis testing. In State of the Art in Probability and Statistics: A Festschrift for Willem R. van Zwet 91--112. IMS, Hayward, CA.
Mathematical Reviews (MathSciNet):
MR1836556
Digital Object Identifier: doi:10.1214/lnms/1215090064
Brillinger, D. R. and Tukey, J. W. (1984). Spectrum analysis in the presence of noise: Some issues and examples. In The Collected Works of John W. Tukey (D. R. Brillinger, ed.) 2 1001--1141. Wadsworth, Monterey, CA.
Zentralblatt MATH:
0699.62084
Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet):
MR1478673
Zentralblatt MATH:
0886.62001
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1--26.
Mathematical Reviews (MathSciNet):
MR515681
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aos/1176344552
Project Euclid: euclid.aos/1176344552
Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.
Mathematical Reviews (MathSciNet):
MR1270903
Zentralblatt MATH:
0835.62038
Fisher, R. A. (1930). Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh.
Zentralblatt MATH:
56.1106.12
Fisher, R. A. (1956). Statistical Methods and Scientific Inference. Hafner, New York.
Zentralblatt MATH:
0070.36903
Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1145237
Knuth, D. E. (1969). The Art of Computer Programming 2. Addison--Wesley, Reading, MA.
Mathematical Reviews (MathSciNet):
MR378456
Le Cam, L. and Yang, G. L. (1990). Asymptotics in Statistics: Some Basic Concepts. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1066869
Zentralblatt MATH:
0719.62003
Mammen, E. (1992). When Does Bootstrap Work? Asymptotic Results and Simulations. Lecture Notes in Statist. 77. Springer, New York.
Zentralblatt MATH:
0760.62038
Miller, R. (1966). Simultaneous Statistical Inference. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet):
MR215441
Zentralblatt MATH:
0192.25702
Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1707286
Quenouille, M. H. (1959). Rapid Statistical Calculations: A Collection of Distribution-Free and Easy Methods of Estimation and Testing. Griffin, London.
Mathematical Reviews (MathSciNet):
MR117806
Zentralblatt MATH:
0087.13904
Savage, L. J. (1954). The Foundations of Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR63582
Zentralblatt MATH:
0055.12604
Student (1908a). The probable error of a mean. Biometrika 6 1--25.
Student (1908b). Probable error of a correlation coefficient. Biometrika 6 302--310.
Tukey, J. W. (1970). Exploratory Data Analysis (limited preliminary edition). Addison--Wesley, Reading, MA. (Regular edition published in 1977.)
Wald, A. (1950). Statistical Decision Functions. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR36976
Zentralblatt MATH:
0040.36402
