• Bernoulli
  • Volume 3, Number 2 (1997), 149-179.

Second-order properties of an extrapolated bootstrap without replacement under weak assumptions

Patrice Bertail

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This paper shows that a straightforward extrapolation of the bootstrap distribution obtained by resampling without replacement, as considered by Politis and Romano, leads to second-order correct confidence intervals, provided that the resampling size is chosen adequately. We assume only that the statistic of interest Tn, suitably renormalized by a regular sequence, is asymptotically pivotal and admits an Edgeworth expansion on some differentiable functions. The results are extended to a corrected version of the moving-block bootstrap without replacement introduced by Künsch for strong-mixing random fields. Moreover, we show that the generalized jackknife or the Richardson extrapolation of such bootstrap distributions, as considered by Bickel and Yahav, leads to better approximations.

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Bernoulli, Volume 3, Number 2 (1997), 149-179.

First available in Project Euclid: 25 April 2007

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bootstrap Edgeworth expansion generalized jackknife random fields Richardson extrapolation strong mixing undersampling


Bertail, Patrice. Second-order properties of an extrapolated bootstrap without replacement under weak assumptions. Bernoulli 3 (1997), no. 2, 149--179.

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  • [1] Babu, G. and Singh, K. (1985) Edgeworth expansions for sampling without replacement from finite populations. J. Multivariate Anal. 17, 261-278.
  • [2] Barbe, P. and Bertail, P. (1995) The Weighted Bootstrap, Lecture Notes in Statist. 98. New York: Springer-Verlag.
  • [3] Beran, R. (1984) Bootstrap methods in statistics. Jahresber. Deutsch. Math. Verein., 86, 24-30.
  • [4] Bertail, P. (1992) La méthode du bootstrap, quelques applications et résultats théoriques. Doctoral thesis, Université de Paris IX.
  • [5] Bertail, P. (1993) Second-order properties of a corrected bootstrap without replacement. Technical Report 9306, INRA-CORELA, Ivry, France.
  • [6] Bertail, P. and Politis, D.N. (1996) Extrapolation of subsampling distribution estimators in i i d and strong-mixing cases. Technical Report 9604, INRA-CORELA, Ivry, France.
  • [7] Bertail, P., Politis, D.N. and Romano, J.P. (1995) On subsampling estimators with unknown rate of convergence. Technical Report 9501, INRA-CORELA, Ivry, France.
  • [8] Bhattacharya, R.N. and Denker, M. (1990) Asymptotic Statistics. Boston: Birkhäuser Verlag.
  • [9] Bhattacharya, R.N. and Ghosh, J. (1978) On the validity of Edgeworth expansion. Ann. Statist., 6, 434-451.
  • [10] Bhattacharya, R.N. and Qumsiyeh, M. (1989) Second-order comparisons between the bootstrap and empirical Edgeworth expansions. Ann. Statist., 17, 160-169.
  • [11] Bickel, P.J. and Freedman, D.A. (1981) Some asymptotic theory for the bootstrap. Ann. Statist., 9, 1196-1217.
  • [12] Bickel, P.J. and Yahav, J.A. (1988) Richardson extrapolation and the bootstrap. J. Amer. Statist. Assoc., 83, 387-393.
  • [13] Bickel, P.J., Götze, F. and van Zwet, W.R. (1994) Resampling fewer than n observations: gains, losses and remedies for losses. Technical report no. 419, University of California.
  • [14] Booth, J.G. and Hall, P. (1993) An improvement of the jackknife distribution function estimator. Ann. Statist., 21, 1476-1485.
  • [15] Bose, A. (1988) Edgeworth corrected by bootstrap in autoregressions. Ann. Statist., 16, 1709-1722.
  • [16] Bosq, D. (1993) Bernstein type large deviation inequalities for partial sums of strong-mixing processes. Statistics, 24, 59-70.
  • [17] Bretagnolle, J. (1983) Lois limites du bootstrap de certaines fonctionelles. Ann. Inst. H. Poincaré Probab. Statist., 19, 281-296.
  • [18] Carlstein, E. (1986) The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Statist., 14, 1171-1179.
  • [19] Chow, Y.S. and Teicher, H. (1988) Probability Theory, Independence, Interchangeability, Martingales, 2nd edn. New York: Springer-Verlag.
  • [20] Doukhan, P. (1994) Mixing: Properties and Examples, Lecture Notes in Statist. 85. New York: Springer-Verlag.
  • [21] Efron, B. (1979) Bootstrap methods: another look at the jackknife. Ann. Statist., 7, 1-26.
  • [22] Falk, M. and Reiss, R.D. (1989) Weak convergence of smoothed and non-smoothed bootstrap quantile estimates. Ann. Probab., 17, 362-371.
  • [23] Götze, F. (1984) Expansions for von mises functionals. Z. Wahrscheinlichkeitstheorie Verw. Geb., 65, 599-625.
  • [24] Götze, F. (1989) Edgeworth expansions in functional limit theorems. Ann. Probab., 17, 1602-1634.
  • [25] Götze, F. and Hipp, C. (1983) Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrscheinlichkeitstheorie Verw. Geb., 64, 211-239.
  • [26] Götze, F. and Hipp, C. (1989) Asymptotic expansions for potential functions of i i d random fields. Probab. Theory Related Fields, 82, 349-370.
  • [27] Götze, F. and Künsch, H.R. (1993) Second-order correctness of the blockwise bootstrap for stationary observations. Preprint 93-061, Sonderforschungsbereich 343 Bielefeld.
  • [28] Gray, H., Schucany, W. and Watkins, T. (1972) The Generalized Jackknife Statistic. New York: Marcel Dekker.
  • [29] Hall, P. (1991a) Edgeworth expansions for nonparametric density estimators, with applications. Statistics, 22, 215-232.
  • [30] Hall, P. (1991b) On convergence rates of suprema. Probab. Theory Related Fields, 89, 447-455.
  • [31] Hall, P. (1992a) Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann. Statist., 20, 675-694.
  • [32] Hall, P. (1992b) The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.
  • [33] Hall, P. and Martin, M. (1991) On the error incurred using the bootstrap variance estimate when constructing confidence intervals. J. Multivariate Anal., 38, 70-81.
  • [34] Isaacson, E. and Keller, H.B. (1966) Analysis of Numerical Methods. New York: Wiley.
  • [35] Künsch, H.R. (1984) Infinitesimal robustness for autoregressive processes. Ann. Statist., 12, 843-863.
  • [36] Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. Ann. Statist., 17, 1217-1241.
  • [37] Lahiri, S.N. (1992) Edgeworth corrected by 'moving-block' bootstrap for stationary and non stationary data. In R. Le Page and L. Billard (eds), Exploring the Limits of the Bootstrap. New York: Wiley.
  • [38] Lo, A.Y. (1991) Bayesian bootstrap clones and a biometry function. Sankhya A, 53, 320-333.
  • [39] Liu, R. and Singh, K. (1992) Moving blocks jackknife and bootstrap capture weak dependence. In R. Le Page and L. Billard (eds), Exploring the Limits of the Bootstrap. New York: Wiley.
  • [40] Maritz, J.S. and Jarrett, R.G. (1978) A note on estimating the variance of the sample median. J. Amer. Statist. Assoc., 82, 155-162.
  • [41] Mason, D. and Newton, M.A. (1992) A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20, 1611-1624.
  • [42] Pfanzagl, J. and Wefelmeier, W. (1985) Asymptotic Expansions for General Statistical Models. New York: Springer-Verlag.
  • [43] Politis, D.N. and Romano, J.P. (1992) A general resampling scheme for triangular arrays of α-mixing random variables with applications to the problem of spectral density estimation. Ann. Statist., 20, 1985-2007.
  • [44] Politis, D.N. and Romano, J.P. (1993) Nonparametric resampling for homogeneous strong-mixing random fields. J. Multivariate Anal., 47, 301-328.
  • [45] Politis, D.N. and Romano, J.P. (1994) A general theory for large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist., 22, 2031-2050.
  • [46] Rhomari, N. (1993) Remarque sur l´inégalité de type exponentielle pour des sommes partielles d´un processus fortement mélangeant. Preprint, LSTA Paris VI and CREST-ENSAE.
  • [47] Sargan, D. (1976) Econometric estimators and the Edgeworth expansion. Econometrica, 44, 421-448.
  • [48] Sargan, D. (1979) Some approximations to the distribution of econometric criteria asymptotically distributed as chi-squared. Econometrica, 49, 1107-1128.
  • [49] Sherman, M. (1992) Subsampling and asymptotic normality for a general statistic from a random field. Ph.D. thesis, Dept. of Statistics, University of North Carolina, Chapel Hill.
  • [50] Shao, J. and Wu, C.F.J. (1989) A general theory for jackknife variance estimation. Ann. Statist., 17, 1176-1197.
  • [51] Tu, D. (1992) Approximating the distribution of a general standardized functional statistic with that of jackknife pseudo-values. In R. Le Page and L. Billard (eds), Exploring the Limits of the Bootstrap. New York: Wiley.
  • [52] Wu, C.F.J. (1990) On the asymptotic properties of the jackknife histogram. Ann. Statist., 18, 1438-1452.