Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-18T22:37:02.507Z Has data issue: false hasContentIssue false
  • English
  • Français

On the accuracy of bootstrapping sample quantiles of strongly mixing sequences

Part of: Nonparametric inference Inference from stochastic processes

Published online by Cambridge University Press:  09 April 2009

Shuxia Sun
Shuxia Sun
Affiliation:
Department of Mathematics and Statistics Wright State University Dayton, OH 45435USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we examine the rate of convergence of moving block bootstrap (MBB) approximations to the distributions of normalized sample quantiles based on strongly mixing observations. Under suitable smoothness and regularity conditions on the one-dimensional marginal distribution function, the rate of convergence of the MBB approximations to distributions of centered and scaled sample quantiles is of order O(n−1¼ log logn).

Keywords

weakly dependentstrictly stationarystrongly mixingmoving block bootstrapsample quantile.

MSC classification

Secondary: 62G09: Resampling methods 62M10: Time series, auto-correlation, regression, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 2 , April 2007 , pp. 263 - 282
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Babu, G. J. and Singh, K., ‘On deviations between empirical and quantile processes for mixing random variables’, J. Multivariate Anal. 8 (1978), 532–549.CrossRefGoogle Scholar
[2]Billingsley, P., Probability and measure (Wiley, New York, 1995).Google Scholar
[3]Bosq, D., Nonparametric statistics for stochastic processes (Springer, New York, 1998).CrossRefGoogle Scholar
[4]Bühlmann, P., ‘Sieve bootstrap for time series’, Bernoulli 3 (1997), 123–148.Google Scholar
[5]Bühlmann, P., ‘Sieve bootstrap with variable-length Markov chains for stationary categorical time series’, J. Amer. Statist. Assoc. 97 (2002), 443–456.CrossRefGoogle Scholar
[6]Bühlmann, P. and Künsch, H. R., ‘Block length selection in the bootstrap for time series’, Comput. Statist. Data Anal. 31 (1999), 295–310.CrossRefGoogle Scholar
[7]Carlstein, E., ‘The use of subseries methods for estimating the variance of a general statistic from a stationary time series’, Ann. Statist. 14 (1986), 1171–1179.CrossRefGoogle Scholar
[8]Efron, B., ‘Bootstrap methods: another look at the jackknife’, Ann. Statist. 7 (1979), 1–26.CrossRefGoogle Scholar
[9]Falk, M. and Janas, J., ‘Edgeworth expansions for studentized and prepivoted sample quantiles’, Statist. Probab. Let. 14 (1992), 13–24.Google Scholar
[10]Falk, M. and Reiss, R. D., ‘Weak convergence of smoothed and nonsmoothed bootstrap quantile estimates’, Ann. Probab. 17 (1989), 362–371.Google Scholar
[11]Götze, F. and Künsch, H. R., ‘Second-order correctness of the blockwise bootstrap for stationary observations’, Ann. Statist. 24 (1996), 1914–1933.CrossRefGoogle Scholar
[12]Hall, P., ‘Resampling a coverage pattern’, Stoch. Proc. Appl. 20 (1985), 231–246.CrossRefGoogle Scholar
[13]Hall, P., Diciccio, T. J. and Romano, J. P., ‘On smoothing and the bootstrap’, Ann. Statist. 17 (1989), 692–704.CrossRefGoogle Scholar
[14]Hall, P., Horowitz, J. L. and Jing, B. Y., ‘On blocking rules for the bootstrap with dependent data’, Biometrika 82 (1995), 561–574.Google Scholar
[15]Künsch, H. R., ‘The jackknife and the bootstrap for general stationary observations’, Ann. Statist. 17 (1989), 1217–1261.Google Scholar
[16]Lahiri, S. N., ‘Second order optimality of stationary bootstrap’, Statist. Probab. Lett. 11 (1991), 335–341.CrossRefGoogle Scholar
[17]Lahiri, S. N., ‘Edgeworth correction by moving block bootstrap for stationary and nonstationary data’, in: Exploring the limits of bootstrap (eds. Lepage, R. and Billard, L.) (Wiley, New York, 1992) pp. 263–270.Google Scholar
[18]Lahiri, S. N., ‘On edgeworth expansion and moving block bootstrap for studentized m-estimators in multiple linear regression models’, J. Multivariate Anal. 56 (1996), 42–59.CrossRefGoogle Scholar
[19]Lahiri, S. N., Resampling methods for dependent data (Springer, New York, 2003).CrossRefGoogle Scholar
[20]Liu, R. Y. and Singh, K., ‘Moving blocks jackknife and bootstrap capture weak convergence’, in: Exploring the limits of bootstrap (eds. Lepage, R. and Billard, L.) (Wiley, New York, 1992) pp. 225–248.Google Scholar
[21]Paparoditis, E. and Politis, D. N., ‘Tapered block bootstrap’, Biometrika 88 (2001), 1105–1119.Google Scholar
[22]Politis, D. and Romano, J. P., ‘A circular block resampling procedure for stationary data’ in: Exploring the limits of bootstrap (eds. Lepage, R. and Billard, L.), (Wiley, New York, 1992) pp. 263–270.Google Scholar
[23]Politis, D. and Romano, J. P., ‘Stationary bootstrap’, J. Amer. Statist. Assoc. 89 (1994), 1303–1313.CrossRefGoogle Scholar
[24]Sen, P. K., ‘On bahadur representation of sample quantile for sequences of φ-mixing random variables’, J. Multivariate Anal. 2 (1972), 77–95.CrossRefGoogle Scholar
[25]Singh, K., ‘On asymptotic accuracy of Efron's bootstrap’, Ann. Statist. 9 (1981), 1187–1195.Google Scholar
[26]Sun, S., Bootstrapping the sample quantile based on weakly dependent observations (Ph.D. Thesis, Department of Statistics, Iowa State University. Ames, IA, 50011, 2004), available at: http://www.wright.edu/~shxia.sun.Google Scholar