Abstract
We show that convergence of intuitive bootstrap distributions to the correct limit distribution is equivalent to a local asymptotic equivariance property of estimators and to an asymptotic independence property in the bootstrap world. The first equivalence implies that bootstrap convergence fails at superefficiency points in the parameter space. However, superefficiency is only a sufficient condition for bootstrap failure. The second equivalence suggests graphical diagnostics for detecting whether or not the intuitive bootstrap is trustworthy in a given data analysis. Both criteria for bootstrap convergence are related to Hájek's (1970, Zeit. Wahrscheinlichkeitsth., 14, 323-330) formulation of the convolution theorem and to Basu's (1955, Sankhyā, 15, 377-380) theorem on the independence of an ancillary statistic and a complete sufficient statistic.
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REFERENCES
Athreya, K. B. (1987). Bootstrap of the mean in the infinite variance case, Ann. Statist., 15, 724–731.
Basu, D. (1955). On statistics independent of a complete sufficient statistic, Sankhyā, 15, 377–380.
Beran, R. (1982). Estimated sampling distributions: the bootstrap and competitors, Ann. Statist., 10, 212–225.
Beran, R. (1984). Bootstrap methods in statistics, Jahresber. Deutsch. Math.-Verein., 86, 212–225.
Beran, R. (1988). Prepivoting test statistics: A bootstrap view of asymptotic refinements, J. Amer. Statist. Assoc., 83, 687–697.
Beran, R. (1995). Stein confidence sets and the bootstrap, Statist. Sinica, 5, 109–127.
Beran, R. and Srivastava, M. S. (1985). Bootstrap tests and confidence regions for functions of a covariance matrix, Ann. Statist., 13, 95–115 (Correction: Ann. Statist., 15, 470–471).
Bickel, P. J. and Freedman, D. F. (1981). Some asymptotic theory for the bootstrap, Ann. Statist., 9, 1196–1217.
Bickel, P. J., Götze, F. and von Zwet, W. R. (1997). Bootstrapping fewer than n observations: gains, losses, and remedies for losses, Statist. Sinica, 7 (in press).
Bretagnolle, J. (1983). Lois limites du bootstrap de certaines fonctionnelles, Ann. Inst. Henri Poincaré., X9, 281–296.
Droste, W. and Wefelmeyer, W. (1984). On Hájek's convolution theorem, Statist. Decisions, 2, 131–144.
Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap, Chapman and Hall, London.
Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates, Ann. Statist., 20, 121–145.
Hájek, J. (1970). A characterization of limiting distributions of regular estimates, Zeit. Wahrscheinlichkeitsth., 14, 323–330.
Hájek, J. (1972). Local asymptotic minimax and admissibility in estimation, Proc. Sixth Berkeley Symp. on Math. Statist. Prob. (eds. L. M. LeCam, J. Neyman, and E. M. Scott), Vol. 1, 175–194, Univ. California Press, Berkeley.
Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests, Academic Press, New York.
Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer, New York.
Hjorth, J. S. U. (1994). Computer Intensive Statistical Methods, Chapman and Hall, London.
Ibragimov, I. A. and Has'minskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory, Springer, New York.
Inagaki, N. (1970). On the limiting distribution of a sequence of estimators with uniformity property, Ann. Inst. Statist. Math., 22, 1–13.
Jeganathan, P. (1981). On a decomposition of the limit distribution of a sequence of estimators, Sankhyā Ser. A, 43(1), 26–36.
Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations, Ann. Statist., 17, 1217–1241.
LeCam, L. (1953). On some asymptotic properties of maximum likelihood estimates and related Bayes estimates, Univ. Calif. Pub. Statist., 1, 277–330.
LeCam, L. (1956). On the asymptotic theory of estimation and testing hypotheses, Proc. Third Berkeley Symp. on Math. Statist. Prob. (ed. J. Neyman), Vol. 1, 129–156, Univ. California Press, Berkeley.
LeCam, L. (1960). Locally asymptotically normal families of distributions, Univ. Calif. Pub. Statist. 3, 27–98.
LeCam, L. (1969). Théorie Asymptotique de la Décision Statistique, Univ. Montreal Press.
LeCam, L. (1973). Sur les contraintes imposées par les passages à limite usuels en statististique, Bull. Internat. Statist. Inst., 45, 169–180.
Lehmann, E. L. (1983). Theory of Point Estimation, Wiley, New York.
Mammen, E. (1992). When Does Bootstrap Work? Lecture Notes in Statist., 77, Springer, New York.
Munroe, M. E. (1953). Introduction to Measure and Integration, Addison-Wesley, Reading, Massachusetts.
Pfanzagl, J. (1994). Parametric Statistical Theory, Walter de Gruyter, Berlin.
Politis, D. N. and Romano, J. (1992). A general resampling scheme for triangular arrays of α-mixing random variables with application to the problem of spectral density estimation, Ann. Statist., 20, 1985–2007.
Putter, H. (1994). Consistency of Resampling Methods, Ph.D. dissertation at Leiden University, The Netherlands.
Swanepool, J. W. II. (1986). A note on proving that the (modified) bootstrap works, Comm. Statist. Theory Methods, 15, 3193–3203.
Titterington, D. M. (1991). Choosing the regularization parameter in image restoration, Spatial Statistics and Imaging (ed. A. Possolo), IMS Lecture Notes-Monograph Series, 20, 392–402.
Welch, B. L. (1937). The significance of the difference between two means when the population variances are unequal, Biometrika, 29, 350–362.
Wu, C. F. J. (1986). Jackknife, bootstrap, and other resampling methods in regression analysis (with discussion), Ann. Statist., 14, 1261–1295.
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Beran, R. Diagnosing Bootstrap Success. Annals of the Institute of Statistical Mathematics 49, 1–24 (1997). https://doi.org/10.1023/A:1003114420352
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DOI: https://doi.org/10.1023/A:1003114420352