A Block Bootstrap Procedure for Long Memory Processes

Argjir Butka; Llukan Puka

doi:https://doi.org/10.14445/22315373/IJMTT-V14P511

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 14 | Number 2 | Year 2014 | Article Id. IJMTT-V14P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V14P511

A Block Bootstrap Procedure for Long Memory Processes

Argjir Butka , Llukan Puka

Citation :

Argjir Butka , Llukan Puka, "A Block Bootstrap Procedure for Long Memory Processes," International Journal of Mathematics Trends and Technology (IJMTT), vol. 14, no. 2, pp. 72-78, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V14P511

Abstract

Long range dependence (LRD) or long memory, initially observed in data analysis in hydrology, is established in diverse fields, for example in financial economics, networks traffic, psychology, cardiology, etc. The presence of long memory in a stochastic process has important consequences in statistical inferences such is a slower rate of decaying of the variance of the sample mean than the typical rate under short dependence. The block bootstrap has been largely developed for weakly dependent time series. Much of research has focused on the large properties of block bootstrap inference about sample means. The block bootstrap for time series consists in randomly resampling blocks of consecutive values of the given data and aligning these blocks into a bootstrap sample. In this paper, we consider a bootstrap technique, which uses blocks composed of cycles, to estimate the variance of the sample mean. The blocks are compounded by a fix number of cycles. The total number of cycles and their lengths are random. The finite sample properties of the method are investigated by means of Monte Carlo experiments and the results indicate that it can be used as an alternative to other block bootstrap methods.

Keywords

Long range dependence, time series, block bootstrap, variance estimation.

References

[1] B. Efron, “Bootstrap methods: another look at the jackknife”, The Annals of statistics, vol. 7, No. 1, pp. 1-26, 1979.
[2] H. Kunsch, “The jackknife and bootstrap for general stationary observations”, The Annals of Statistics, vol. 17, No. 3, pp. 1217-1241, 1989.
[3] R. Liu and K. Singh, “Moving blocks jackknife and bootstrap capture weak dependence”, Exploring the Limits of Bootstrap, Wiley, New York, pp. 225-248, 1992.
[4] E. Carlstein, “The use of subseries values for estimating the variance of a general statistic from a stationary sequence”, The Annals of Statistics, vol. 14, No. 3, pp. 1171-1179, 1986.
[5] S. N. Lahiri, “On the moving block bootstrap under long range dependence”, Statistics & Probability Letters, vol. 18, pp. 405-413, 1993.
[6] E. Carlstein, K. Do, P. Hall, T. Hesterberg, and H. R. Kunsch, “Matched-block bootstrap for dependent data”, Bernoulli, vol. 4, No. 3, pp. 305-328, 1998.
[7] G.E.P. Box and G.M. Jenkins, Time series analysis, forecasting and control. Oakland, California: Holden-Day, 1976.
[8] P. Brockwell and R. Davis, Time series: theory and methods. Springer-Verlag, 1991.
[9] A. I. McLeod and K.W. Hipel, “Preservation of the Rescaled Adjusted Range 1. A reassessment of the Hurst Phenomenon”, Water Resources Research, vol. 14, No. 3, pp. 491-508, 1978.
[10] J. Beran, Statistics for long-memory processes, New York: Chapman & Hall, 1994.
[11] C. W. Granger and R. Joyeux, “An introduction to long memory time series models and fractional differencing”, Journal of Time Series Analysis, vol. 1, pp. 15–39, 1980.
[12] J. Hosking, “Fractional differencing”, Biometrika, vol. 68, No. 1, pp. 165-176, 1981.
[13] B. B. Mandelbrot, and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications”, SIAM Review, vol. 10, No. 4, pp. 422-437, 1968.
[14] D. Guegan, “How can we Define the Concept of Long Memory? An Econometric Survey”, Econometric Reviews, vol. 24, No. 2, pp. 113-149, 2005.
[15] P. Hall, “Resampling a coverage pattern”, Stochastic Processes and their Applications, vol. 20, pp. 231-246, 1985.
[16] D.N. Politis and J. Romano, “Circular block resampling procedure for stationary data”, Exploring the Limits of Bootstrap, Wiley-New York, pp. 263-270, 1992.
[17] D. Politis and J. Romano, “The stationary bootstrap”, JASA, vol. 89, pp. 1303-1313, 1994.
[18] G.C. Franco and V.A. Reisen, “Bootstrap techniques in semiparametric estimation methods for ARFIMA models: a comparison study”, Computational Statistics & Data Analysis, vol. 19, pp. 243–259, 2004.
[19] T. Hesterberg, “Matched-Block Bootstrap for Long Memory Processes”, MathSoft, Inc., Seattle, USA, Research Report No. 66, 1997.
[20] P. Hall, J. L. Horowitz and B-Y. Jing, “On blocking rules for the bootstrap with dependent data”, Biometrika, vol. 82, No. 3, pp. 561-574, 1995.
[21] D.N. Politis, and H. White “Automatic block-length selection for the dependent bootstrap”, Econometric Reviews, vol. 23, No.1, pp. 53-70, 2004.
[22] A. Patton, D.N. Politis and H. White, “CORRECTION TO "Automatic block-length selection for the dependent bootstrap" by D. Politis and H. White”, Econometric Reviews, vol. 28, No. 4, pp. 372-375, 2009.
[23] Ekonomi, L. and Butka, A. 2011. “Jackknife and bootstrap with cycling blocks for the estimation of fractional parameter in ARFIMA model”, Turk. J. Math., 35(1), 151-158.
[24] A. Butka, L. Puka and I. Palla, “Bootstrap testing for long range dependence”, International Journal of Mathematics Trends and Technology, vol. 8, No. 3, pp. 164-172, 2014.
[25] D. Nordman and S.N. Lahiri, “Validity of the sampling window method for long-range dependent linear processes”, Econometric Theory, vol. 21, pp. 1087-1111, 2005.
[26] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, 1993.