Abstract
The block bootstrap has been largely developed for weakly dependent time processes and, in this context, much research has focused on the large-sample properties of block bootstrap inference about sample means. This work validates the block bootstrap for distribution estimation with stationary, linear processes exhibiting strong dependence. For estimating the sample mean’s variance under long-memory, explicit expressions are also provided for the bias and variance of moving and non-overlapping block bootstrap estimators. These differ critically from the weak dependence setting and optimal blocks decrease in size as the strong dependence increases. The findings in distribution and variance estimation are then illustrated using simulation.
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References
Adenstedt, R.K. (1974). On large-sample estimation for the mean of a stationary sequence. Ann. Statist., 2, 1095–1107.
Andrews D.W.K. and Sun, Y. (2004). Adaptive local polynomial Whittle estimation of long-range dependence. Econometrica, 72, 569–614.
Andrews, D.W.K., Lieberman, O. and Marmer, V. (2006). Higher-Order Improvements of the Parametric Bootstrap for Long-memory Gaussian processes. J. Econometrics, 133, 673–702.
Beran, J. (1994). Statistical methods for long memory processes. Chapman & Hall, London.
Brockwell, P.J. and Davis, R.A. (1991). Time series: theory and methods. Second Edition, Springer, New York.
Buhlmann, P. (1997). Sieve bootstrap for time series. Bernoulli, 3, 123–148.
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.
Choi, E. and Hall, P. (2000). Bootstrap confidence regions constructed from autoregressions of arbitrary order. J. R. Stat. Soc., Ser. B, 62, 461–477.
Davydov, Y.A. (1970). The invariance principle for stationary processes. Theor. Prob. Appl., 15, 487–498.
Efron, B. (1979). Bootstrap methods: another look at the jackknife. Ann. Statist., 7, 1–26.
Fay, G., Moulines, E. and Soulier, P. (2004). Edgeworth expansions for linear statistics of possibly long-range-dependent linear processes. Statist. Probab. Lett., 66, 275–288.
Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of longmemory time series models. J. Time Series Anal., 4, 221–238.
Giraitis, L., Robinson, P.M. and Surgailis, D. (1999). Variance-type estimation of long memory. Stochastic Process. Appl., 80, 1–24.
Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Series Anal., 1, 15–29.
Hall, P., Horowitz, J.L. and Jing, B.-Y. (1996). On blocking rules for the bootstrap with dependent data. Biometrika, 82, 561–574.
Hall, P., Jing, B.-Y. and Lahiri, S.N. (1998). On the sampling window method for long-range dependent data. Statist. Sinica, 8, 1189–1204.
Hidalgo, J. (2003). An alternative bootstrap to moving blocks for time series regression models. J. Econometrics, 117, 369–399.
Hosking, J.R.M. (1981). Fractional differencing. Biometrika, 68, 165–176.
Hurvich, C.M., Deo, R. and Brodsky, J. (1998). The mean square error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long memory time series. J. Time Series Anal., 19, 19–46.
Ibragimov, I. A. and Linnik, Y.V. (1971). Independent and stationary sequences of random variables. Wolters-Noordhoff, Groningen.
Kapetanios, G. and Psaradakis, Z. (2006). Sieve bootstrap for strongly dependent stationary processes. Working paper 552, Dept. of Economics, Queen Mary, University of London.
Kreiss, J.P. and Paparoditis, E. (2003). Autogressive aided periodogram bootstrap for time series. Ann. Statist., 31, 1923–1955.
Künsch, H.R. (1987). Statistical aspects of self-similar processes. In Proceedings of the 1st World Congress of the Bernoulli Society, 1, (Yu. A. Prohorov and V. V, Sazanov, eds.), 67–74. VNU Science Press, Utrecht.
Künsch, H.R. (1989). The jackknife and bootstrap for general stationary observations. Ann. Statist., 17, 1217–1261.
Lahiri, S.N. (1993). On the moving block bootstrap under long range dependence. Statist. Probab. Lett., 18, 405–413.
Lahiri, S.N. (1999). Theoretical comparisons of block bootstrap methods. Ann. Statist., 27, 386–404.
Lahiri, S.N. (2003). Resampling methods for dependent data. Springer, New York.
Lahiri, S.N., Furukawa, K. and Lee, Y-D. (2007). A nonparametric plug-in rule for selecting optimal block lengths for block bootstrap methods. Stat. Methodol., 4, 292–321.
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. John Wiley & Sons, New York.
Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev., 10, 422–437.
McElroy, T. and Politis, D.N. (2007). Self-normalization for heavy-tailed time series with long memory. Statist. Sinica, 17, 199–220.
Moulines, E. and Soulier, P. (2003). Semiparametric spectral estimation for fractional processes. In P. Doukhan, G. Oppenheim and M. S. Taqqu (eds.), Theory and Applications of Long-range Dependence, (P. Doukhan, G. Oppenheim and M. S. Taqqu, eds.), 251–301. Birkhäuser, Boston.
Nordman, D.J. and Lahiri, S.N. (2005). Validity of the sampling window method for long-range dependent linear processes. Econometric Theory, 21, 1087–1111.
Politis, D.N. and White, H. (2004). Automatic block-length selection for the dependent bootstrap. Econometric Rev., 23, 53–70.
Poskitt, D.S. (2007). Properties of the sieve bootstrap for fractionally integrated and non-invertible processes. J. Time Series Anal., 29, 224–250.
Robinson, P. M. (1995a). Gaussian semiparametric estimation of long range dependence. Ann. Statist., 23, 1630–1661.
Robinson, P. M. (1995b). Log-periodogram regression of time series with long range dependence. Ann. Statist., 23, 1048–1072.
Singh, K. (1981). On the asymptotic accuracy of the Efron’s bootstrap. Ann. Statist., 9, 1187–1195.
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Kim, Y.M., Nordman, D.J. Properties of a block bootstrap under long-range dependence. Sankhya A 73, 79–109 (2011). https://doi.org/10.1007/s13171-011-0003-3
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DOI: https://doi.org/10.1007/s13171-011-0003-3