# zbMATH — the first resource for mathematics

Bootstrapping the empirical distribution of a stationary process with change-point. (English) Zbl 1431.62180
Authors’ abstract: When detecting a change-point in the marginal distribution of a stationary time series, bootstrap techniques are required to determine critical values for the tests when the pre-change distribution is unknown. In this paper, we propose a sequential moving block bootstrap and demonstrate its validity under a converging alternative. Furthermore, we demonstrate that power is still achieved by the bootstrap under a non-converging alternative. We follow the approach taken by M. Peligrad [Ann. Probab. 26, No. 2, 877–901 (1998; Zbl 0932.62055)], and avoid assumptions of mixing, association or near epoch dependence. These results are applied to a linear process and are shown to be valid under very mild conditions on the existence of any moment of the innovations and a corresponding condition of summability of the coefficients.

##### MSC:
 62G09 Nonparametric statistical resampling methods 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G10 Nonparametric hypothesis testing 62G30 Order statistics; empirical distribution functions 60F17 Functional limit theorems; invariance principles 60G10 Stationary stochastic processes
Full Text:
##### References:
 [1] Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications., The Annals of Mathematical Statistics5 1656-1670. · Zbl 0265.60011 [2] Billingsley, P. (1968)., Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201 [3] Doukhan, P. and Surgailis, D. (1998). Functional central limit theorem for the empirical process of short memory linear processes., C.R. Acad. Sci. Paris326 87-92. · Zbl 0948.60012 [4] El Ktaibi, F. (2015). Asymptotics for the sequential empirical process and testing for distributional change for stationary linear models. Ph.D. thesis, University of Ottawa., http://hdl.handle.net/10393/31916 [5] El Ktaibi, F. and Ivanoff, B. G. (2016) Change-point detection in the marginal distribution of a linear process., Electronic Journal of Statistics10 3945-3985. · Zbl 1353.62095 [6] El Ktaibi, F., Ivanoff, B. G. and Weber, N.C. (2014). Bootstrapping the empirical distribution of a linear process., Statistics and Probability Letters93 134-142. · Zbl 06327901 [7] Giraitis, L., Leipus, R. and Surgailis, D. (1996). The change-point problem for dependent observations., Journal of Statistical Planning and Inference53 297-310. · Zbl 0856.62073 [8] Inoue, A. (2001). Testing for the distributional change in time series., Econometric Theory17 156-187. · Zbl 0976.62088 [9] Ivanoff, B. G. (1980). The function space $$D([0,\infty )^q,E)$$., The Canadian Journal of Statistics8 179-191. · Zbl 0467.60009 [10] Ivanoff, B. G. and Weber, N. C. (2010). Asymptotic results for spatial causal ARMA models., Electronic Journal of Statistics4 15-35. · Zbl 1298.62075 [11] Künsch, H. R. (1989) The jackknife and the bootstrap for general stationary observations., Ann. Stat.17 1217-1241. · Zbl 0684.62035 [12] Lahiri, S. N. (2003)., Resampling Methods for Dependent Data. Springer, Berlin. · Zbl 1028.62002 [13] Naik-Nimbalkar, U. V. and Rajarshi, M. B. (1994). Validity of blockwise bootstrap for empirical processes with stationary observations., Ann. Statis.22 (2) 980-994. · Zbl 0808.62043 [14] Peligrad, M. (1998). On the blockwise bootstrap for empirical processes for stationary sequences., The Annals of Probability26(2) 877-901. · Zbl 0932.62055 [15] Pollard, D. (1984)., Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045 [16] Radulović, D. (2009). Another look at the disjoint blocks bootstrap., Test18, 195-212. · Zbl 1203.62073 [17] Sharipov, O., Tewes, J. and Wendler, M. (2016). Sequential block bootstrap in a Hilbert space with application to change point analysis., Canadian Journal of Statistics44, 300-322. · Zbl 1357.62187 [18] van der Vaart, A. and Wellner, J. (1996)., Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.