Hostname: page-component-77c78cf97d-xcx4r Total loading time: 0 Render date: 2026-04-24T00:16:55.774Z Has data issue: false hasContentIssue false
  • English
  • Français

BLOCK BOOTSTRAP CONSISTENCY UNDER WEAK ASSUMPTIONS

Published online by Cambridge University Press:  01 February 2018

Gray Calhoun
Gray Calhoun*
Affiliation:
Iowa State University
*
*Address correspondence to Gray Calhoun, Economics Department, Iowa State University, Ames, IA 50011, USA; e-mail: gcalhoun@iastate.edu, url: http://gray.clhn.org.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e., the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent (NED) functions of mixing processes; they are consistent under the weakest conditions that ensure the original NED process obeys a central limit theorem (CLT), established by De Jong (1997, Econometric Theory 13(3), 353–367). In doing so, this paper extends De Jong’s method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a functional CLT (FCLT) under the same conditions.

Information

Type
MISCELLANEA
Information
Econometric Theory , Volume 34 , Issue 6 , December 2018 , pp. 1383 - 1406
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

I would like to thank Helle Bunzel, Dimitris Politis, Robert Taylor, and three anonymous referees for their comments and feedback on earlier versions of this paper.

References

REFERENCES

Bai, J. (1994) Least squares estimation of a shift in linear processes. Journal of Time Series Analysis 15(5), 453–472.CrossRefGoogle Scholar
Chow, Y., Robbins, H., & Teicher, H. (1965) Moments of randomly stopped sums. The Annals of Mathematical Statistics 36(3), 789–799.Google Scholar
Dahlhaus, R. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 379–381.Google Scholar
Davidson, J. (1993) An L 1-convergence theorem for heterogeneous mixingale arrays with trending moments. Statistics & Probability Letters 16(4), 301–304.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory: An Introduction for Econometricians. Advanced Texts in Econometrics. Oxford University Press.CrossRefGoogle Scholar
De Jong, R.M. (1997) Central limit theorems for dependent heterogeneous random variables. Econometric Theory 13(3), 353–367.CrossRefGoogle Scholar
De Jong, R.M. & Davidson, J. (2000) The functional central limit theorem and weak convergence to stochastic integrals I: Weakly dependent processes. Econometric Theory 16(5), 621–642.CrossRefGoogle Scholar
Dowla, A., Paparoditis, E., & Politis, D.N. (2003) Locally stationary processes and the Local Block Bootstrap. In Akritas, M.G. & Politis, D.N. (eds.), Recent Advances and Trends in Nonparametric Statistics, pp. 437–444. Elsevier (North Holland).Google Scholar
Gonçalves, S. & De Jong, R.M. (2003) Consistency of the stationary bootstrap under weak moment conditions. Economics Letters 81(2), 273–278.Google Scholar
Gonçalves, S. & Politis, D. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 383–386.CrossRefGoogle Scholar
Gonçalves, S. & White, H. (2002) The bootstrap of the mean for dependent heterogeneous arrays. Econometric Theory 18(6), 1367–1384.CrossRefGoogle Scholar
Hall, P. & Heyde, C. (1980) Martingale Limit Theory and its Application. Academic Press.Google Scholar
Horowitz, J.L. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 387–389.CrossRefGoogle Scholar
Jentsch, C. & Mammen, E. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 391–392.Google Scholar
Kreiss, J.-P. & Paparoditis, E. (2011a) Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 357–378.CrossRefGoogle Scholar
Kreiss, J.-P. & Paparoditis, E. (2011b) Rejoinder: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 393–395.Google Scholar
Kunsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17(3), 1217–1241.Google Scholar
Lahiri, S. (1999) Theoretical comparisons of block bootstrap methods. The Annals of Statistics 27(1), 386–404.CrossRefGoogle Scholar
Liu, R.Y. & Singh, K. (1992) Moving blocks Jackknife and Bootsrap capture weak dependence. In LePage, R. & Billard, L. (eds.), Exploring the Limits of Bootstrap, pp. 225–248. John Wiley.Google Scholar
McLeish, D. (1975) Invariance principles for dependent variables. Probability Theory and Related Fields 32, 165–178.Google Scholar
McLeish, D. (1977) On the invariance principle for nonstationary mixingales. The Annals of Probability 5(4), 616–621.Google Scholar
Nordman, D. (2009) A note on the stationary bootstrap’s variance. The Annals of Statistics 37(1), 359–370.Google Scholar
Paparoditis, E. & Politis, D. (2002) Local block bootstrap. Comptes Rendus Mathematique 335(11), 959–962.Google Scholar
Politis, D.N. & Romano, J.P. (1992) A circular block resampling procedure for stationary data. In Page, R. & LePage, R. (eds.), Exploring the Limits of Bootstrap, pp. 263–270. John Wiley.Google Scholar
Politis, D.N. & Romano, J.P. (1994) The stationary bootstrap. Journal of the American Statistical Association 89(428), 1303–1313.CrossRefGoogle Scholar
Radulović, D. (1996) The bootstrap of the mean for strong mixing sequences under minimal conditions. Statistics & Probability Letters 28(1), 65–72.CrossRefGoogle Scholar
Van der Vaart, A.W. (2000) Asymptotic Statistics. Cambridge University Press.Google Scholar
Supplementary material: PDF

Calhoun supplementary material

Appendix

Download Calhoun supplementary material(PDF)
PDF 319.6 KB
Supplementary material: File

Calhoun supplementary material

Appendix

Download Calhoun supplementary material(File)
File 21 KB