Abstract
In this chapter, we discuss various resampling procedures, such as bootstrap, jackknife, and sample re-use procedures for discrete time stochastic processes. Our discussion begins with bootstrap procedures for finite and infinite Markov chains. Further, bootstrap for stationary real valued Markov sequences based on transition density estimators is discussed. This is followed by bootstrap based on residuals for stationary and invertible linear ARMA time series. We then describe bootstrap and jackknife procedures based on blocks of stationary observations. Further, we discuss a bootstrap procedure based on AR-sieves. Results which prove superiority of block-based bootstrap over the traditional central limit theorems are discussed herein. The last section discusses bootstrap procedures for construction of confidence intervals based on estimating functions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Allen, M., Datta, S.: A note on bootstrapping M-estimators in ARMA models. J. Time Ser. Anal. 20, 365–379 (1999)
Athreya, K.B., Fuh, C.D.: Bootstrapping Markov chains : countable case. J. Stat. Plan. Inf. 33, 311–331 (1992a)
Athreya, K.B., Fuh, C.D.: Bootstrapping Markov chains : countable case. In: LePage, R., Billard, L. (eds.) Exploring the Limits of Bootstrap, pp. 49–60. Wiley, New York (1992b)
Basawa, I.V., Mallik, A.W., McCormick, W.P., Taylor, R.L.: Asymptotic bootstrap validity for finite Markov chains. Commun. Stat. Theor. Meth. 19, 1493–1510 (1990)
Bickel, P.J., Bühlmann, P.: A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5, 413–446 (1999)
Bose, A.: Edgeworth correction by bootstrap in autoregression. Ann. Stat. 16, 1709–1722 (1988)
Bose, A.: Bootstrap in moving average models Ann. Inst. Stat. Math. 42, 753–768 (1990)
Brockwell, P.J., Davis, R.A.: Time Series : Theory and Methods. Springer, New York (1987)
Bühlmann, P.: Blockwise bootstrapped empirical process for stationary sequences. Ann. Stat. 22, 995–1012 (1994)
Bühlmann, P.: Bootstraps for time series. Stat. Sci. 17, 52–72 (2002)
Carlstein, E.: The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Stat. 14, 1172–1179 (1986)
Chatterjee, S., Bose, A.: Generalized bootstrap for estimating equations. Ann. Stat. 33, 414–436 (2005)
Chernick, M.R.: Bootstrap Methods : A Guide for Practitioners and Researchers, 2nd edn. Wiley, New York (2008)
Davison, A.C., Hall, P.: On Studentizing and blocking methods for implementing the bootstrap with dependent data. Aust. J. Stat. 35, 215–224 (1993)
Davison, A.C., Hinkley, D.V.: Bootstrap Methods and Their Application. Cambridge University Press, Cambridge (1997)
Efron, B., Tibshirani, R.: An Introduction to the Bootstrap. Chapman and Hall, London (1993)
Fernholz, L.T.: Von Mises Calculus for Statistical Functionals. Lecture Notes in Statistics 19. Springer, New York (1983)
Freedman, D.A., Peters, S.C.: Bootstrapping a regression equation : some empirical results. J. Amer. Stat. Assoc. 79, 97–106 (1984)
Götze, F., Hipp, C.: Asymptotic expansions for sums of weakly dependent random variables. Z. Wahr. verw. Geb. 64, 211–240 (1983)
Götze, F., Künsch, H.R.: Second-order correctness of the block-wise bootstrap for stationary observations. Ann. Stat. 24, 1914–1933 (1996)
Hall, P., Jing, B.: On sample reuse methods for dependent data. J. Roy. Stat. Soc.: Ser. B 56, 727–738 (1996)
Hall, P., Horowitz, J.L., Jing, B.: On blocking rules for the bootstrap with dependent data. Biometrika 82, 561–574 (1995)
Hu, F.: Efficiency and robustness of a resampling M-estimator in the linear model. J. Multivar. Anal. 78, 252–271 (2001)
Hu, F., Kalbfleisch, J.D.: The estimating function bootstrap (with discussion). Can. J. Stat. 28, 449–499 (2000)
Hu, F., Zidec, J.V.: A bootstrap based on the estimating equations of the linear model. Biometrika 82, 263–275 (1995)
Kreiss, J.P., Franke, J.: Bootstrapping stationary auto-regressive moving-average models. J. Time Ser. Anal. 13, 297–317 (1992)
Kulperger, R.J., Prakasa Rao, B.L.S.: Bootstrapping a finite Markov chain. Sankhyā A 51, 178–191 (1989)
Künsch, H.R.: The jackknife and the bootstrap for general stationary observations. Ann. Stat. 17, 1217–1261 (1989)
Lahiri, S.N.: Second order optimality of stationary bootstrap. Stat. Probab. Lett. 11, 335–341 (1991)
Lahiri, S.N.: Resampling Methods for Dependent Data. Springer, New York (2003)
Lele, S.R.: Jackknifing linear estimating equations: asymptotic theory and applications in stochastic processes. J. Roy. Statist. Soc.: Ser. B 53, 253–267 (1991a)
Lele, S.R.: Resampling using estimating functions. In: Godambe, V.P. (ed.) Estimating Functions, pp. 295–304. Oxford University Press, Oxford (1991b)
Liu, R.Y., Singh, K.: Moving blocks jackknife and bootstrap capture weak dependence. In: LePage, R., Billard, L. (eds.) Exploring the Limits of Bootstrap, pp. 225–248. Wiley, New York (1992)
Naik-Nimbalkar, U.V., Rajarshi, M.B.: Validity of blockwise bootstrap for empirical processes with stationary observations. Ann. Stat. 22, 980–994 (1994)
Paparoditis, E., Politis, D.N.: The local bootstrap for Markov processes. J. Stat. Plan. Inf. 108, 301–328 (2002)
Parr, W.C.: The bootstrap: some large sample theory and connections with robustness. Stat. Probab. Lett. 3, 97–100 (1985)
Politis, D.N., Romano, J.P.: A circular bootstrap resampling procedure for stationary data. In: LePage, R., Billard, L. (eds.) Exploring the Limits of Bootstrap, pp. 263–270. Wiley, New York (1992)
Politis, D.N., Romano, J.P.: The stationary bootstrap. J. Amer. Stat. Assoc. 89, 1303–1313 (1994)
Radulović, D.: On the bootstrap and the empirical processes for dependent sequences. In: Dehling, H., Mikosch, T., Sørensen, M. (eds.) Empirical Process Techniques for Dependent Data, pp. 345–364. Birkhäuser, Boston (2002)
Rajarshi, M.B.: Bootstrap in Markov sequences based on estimates of transition density. Ann. Inst. Stat. Math. 42, 253–268 (1990)
Rao, C.R., Zhao, L.C.: Approximation to the distribution of M-estimates in linear models by randomly weighted bootstrap. Sankhyā A 54, 323–331 (1992)
Rohan, N., Ramanathan, T.V.: Order section in ARMA models using the focused information criterion. Aust. NZ. J. Stat. 53, 217–231 (2011)
Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)
Shao, J., Tu, D.: The Jackknife and Bootstrap. Springer, New York (1995)
Singh, K.: On the asymptotic accuracy of Efron’s bootstrap Ann. Stat. 9, 1187–1195 (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 The Author(s)
About this chapter
Cite this chapter
Rajarshi, M.B. (2013). Bootstrap and Other Resampling Procedures. In: Statistical Inference for Discrete Time Stochastic Processes. SpringerBriefs in Statistics. Springer, India. https://doi.org/10.1007/978-81-322-0763-4_6
Download citation
DOI: https://doi.org/10.1007/978-81-322-0763-4_6
Published:
Publisher Name: Springer, India
Print ISBN: 978-81-322-0762-7
Online ISBN: 978-81-322-0763-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)