The Annals of Statistics

Theoretical comparisons of block bootstrap methods

S. N. Lahiri

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Abstract

In this paper, we compare the asymptotic behavior of some common block bootstrap methods based on nonrandom as well as random block lengths. It is shown that, asymptotically, bootstrap estimators derived using any of the methods considered in the paper have the same amount of bias to the first order. However, the variances of these bootstrap estimators may be different even in the first order. Expansions for the bias, the variance and the mean-squared error of different block bootstrap variance estimators are obtained. It follows from these expansions that using overlapping blocks is to be preferred over nonoverlapping blocks and that using random block lengths typically leads to mean-squared errors larger than those for nonrandom block lengths.

Article information

Source
Ann. Statist., Volume 27, Number 1 (1999), 386-404.

Dates
First available in Project Euclid: 5 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1018031117

Digital Object Identifier
doi:10.1214/aos/1018031117

Mathematical Reviews number (MathSciNet)
MR1701117

Zentralblatt MATH identifier
0945.62049

Subjects
Primary: 62G05: Estimation
Secondary: 62E05

Keywords
Block bootstrap mean squared error stationary bootstrap

Citation

Lahiri, S. N. Theoretical comparisons of block bootstrap methods. Ann. Statist. 27 (1999), no. 1, 386--404. doi:10.1214/aos/1018031117. https://projecteuclid.org/euclid.aos/1018031117


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References

  • Bartlett, M. S. (1946). On the theoretical specification of sampling properties of autocorrelated time series. J. Roy. Statist. Soc. Suppl. 8 27-41.
  • Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451.
  • B ¨uhlman, P. (1994). Blockwise bootstrapped empirical process for stationary sequences. Ann. Statist. 22 995-1012.
  • B ¨uhlman, P. and K ¨unsch, H. R. (1994). Block length selection in the bootstrap for time series. Research Report 72, Seminar f ¨ur Statistik, ETH, Z ¨urich.
  • Bustos, O. (1982). General M-estimates for contaminated pth order autoregressive processes: consistency and asymptotic normality. Z. Wahrsch. Verw. Gebiete 59 491-504.
  • Carlstein, E. (1986). The use of subseries methods for estimating the variance of a general statistic from a stationary time series. Ann. Statist. 14 1171-1179.
  • Carlstein, E., Do, K-A., Hall, P., Hesterberg, T. and K ¨unsch, H. R. (1995). Matched-block bootstrap for dependent data. Research Report 74, Seminar f ¨ur Statistik, ETH, Z ¨urich.
  • Davison, A. C. and Hall, P. (1993). On Studentizing and blocking methods for implementing the bootstrap with dependent data. Austral. J. Statist. 35 215-224.
  • G ¨otze, F. and K ¨unsch, H. R. (1996). Blockwise bootstrap for dependent observations: higher order approximations for Studentized statistics. Ann. Statist. 24 1914-1933.
  • Hall, P. (1985). Resampling a coverage pattern. Stochastic Process. Appl. 20 231-246.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • Hall, P., Horowitz, J. L. and Jing, B. (1995). On blocking rules for the bootstrap with dependent data. Biometrika 82 561-574.
  • Ibragimov, I. A. and Hasminskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, New York.
  • K ¨unsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217-1261.
  • Lahiri, S. N. (1991). Second order optimality of stationary bootstrap. Statist. Probab. Lett. 11 335-341.
  • Lahiri, S. N. (1995). On the asymptotic behaviour of the moving block bootstrap for normalized sums of heavy-tail random variables. Ann. Statist. 23 1331-1349. Lahiri, S. N. (1996a). On Edgeworth expansion and moving block bootstrap for Studentized Mestimators in multiple linear regression models. J. Multivariance Analysis. 56 42-59. Lahiri, S. N. (1996b). On empirical choice of the optimal block size for block bootstrap methods. Preprint. Dept. Statistics, Iowa State Univ., Ames.
  • Lahiri, S. N. (1997). On second-order properties of the stationary bootstrap method for Studentized statistics. Preprint. Dept. Statistics, Iowa State Univ., Ames.
  • Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (R. Lepage and L. Billard, eds.) 225-248. Wiley, New York.
  • Naik-Nimbalkar, U. V. and Rajarshi, M. B. (1994). Validity of blockwise bootstrap for empirical processes with stationary observations. Ann. Statist. 22 980-994.
  • Politis, D. and Romano, J. P. (1992). A circular block resampling procedure for stationary data. In Exploring the Limits of Bootstrap (R. Lepage and L. Billard, eds.) 263-270. Wiley, New York.
  • Politis, D. and Romano, J. P. (1994). The stationary bootstrap. J. Amer. Statist. Assoc. 89 1303- 1313.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series 1. Academic Press, New York.
  • Rudin, W. (1985). Real and Complex Analysis. McGraw-Hill, New York.
  • Shao, Q. M. and Yu, H. (1993). Bootstrapping the sample means for stationary mixing sequences. Stochastic Process. Appl. 48 175-190.
  • Woodroofe, M. B. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia.