Bernoulli

  • 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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 25 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1177526727

Mathematical Reviews number (MathSciNet)
MR1466305

Zentralblatt MATH identifier
0919.62035

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

Citation

Bertail, Patrice. Second-order properties of an extrapolated bootstrap without replacement under weak assumptions. Bernoulli 3 (1997), no. 2, 149--179. https://projecteuclid.org/euclid.bj/1177526727


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References

  • [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.