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Consistency of the frequency domain bootstrap for differentiable functionals. (English) Zbl 1459.62176

The classical frequency domain bootstrap (FDB) is investigated and analyzed herein. First some well-known bootstrap consistency results for spectral density functions are revisited and then the article focuses on smooth functions of linear functionals by providing the necessary and sufficient conditions for the validity of the FDB. The authors establish that the FDB is asymptotically valid if and only if the kurtosis of the process is 0 (for instance in the Gaussian case) or if the functional of interest has a centered influence function, which is the case for ratio statistics as well as some Whittle estimators. Then the conditions under which the empirical process in the frequency domain converges to some Gaussian process and when its bootstrap version is valid are provided. Hence it becomes feasible to prove the validity of the bootstrap for large classes of interesting statistics.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference
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[1] ABRAMOVITCH, L. and SINGH, K. (1985). Edgeworth corrected pivotal statistics and the bootstrap., Ann. Statist. 13 116-132. · Zbl 0575.62018 · doi:10.1214/aos/1176346580
[2] BARBE, P. and BERTAIL, P. (1995)., The Weighted Bootstrap. Lecture Notes in Statistics, Springer N.Y. · Zbl 0826.62030
[3] BERTAIL, P. and CLEMENCON, S. (2006). Regenerative block bootstrap for Markov chains., Bernoulli 12 689-712. · Zbl 1125.62037 · doi:10.3150/bj/1155735932
[4] BERTAIL, P. and CLEMENCON, S. (2007). Approximate regenerative block-bootstrap for Markov chains., Computational Statistics and Data Analysis 52 2739-2756. · Zbl 1452.62031 · doi:10.1016/j.csda.2007.10.014
[5] CHAN, V., LAHIRI, S.N. and MEEKER, W.Q. (2004). Block bootstrap estimation of the distribution of cumulative outdoor degradation., Technometrics 46 215-224.
[6] COATES, D. and DIGGLE, P. (1986). Tests for comparing two estimated spectral densities., Journal of Time Series Analysis 7 7-20. · Zbl 0581.62076 · doi:10.1111/j.1467-9892.1986.tb00482.x
[7] DAHLHAUS, R. (1985). Asymptotic Normality of Spectral Estimates., J. Multivariate Anal. 16 412-431. · Zbl 0579.62082 · doi:10.1016/0047-259X(85)90028-4
[8] DAHLHAUS, R. (1988). Empirical spectral processes and their applications to time series analysis., Stoch. Proc. Appl. 30 69-83. · Zbl 0655.60033 · doi:10.1016/0304-4149(88)90076-2
[9] DAHLHAUS, R. and POLONIK, W. (2002). Empirical Spectral Processes and Nonparametric Maximum Likelihood Estimation for Time Series. In: Dehling H., Mikosch T., Sørensen M. (eds), Empirical Process Techniques for Dependent Data. Birkhäuser, Boston, MA · Zbl 1022.62091
[10] DAHLHAUS, R. and JANAS, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis., Ann. of Stat. 24 1934-1963. · Zbl 0867.62072 · doi:10.1214/aos/1069362304
[11] DETTE, H. and HILDEBRANDT, T. (2012). A note on testing hypotheses for stationary processes in the frequency domain, Journal of Multivariate Analysis 104 101-114. · Zbl 1236.62101 · doi:10.1016/j.jmva.2011.07.002
[12] DUDLEY, R.M. (1990). Nonlinear functionals of empirical measures and the bootstrap. In:, E. Eberlein, J. Kuelbs and M.B. Marcus, eds., Probability in Banach Spaces, Vol. 7, pp. 63-82, Birkhäuser, Boston. · Zbl 0704.62043
[13] DUDEK, A.E. (2015). Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series., Metrika 78(3) 313-335. · Zbl 1333.62209 · doi:10.1007/s00184-014-0505-9
[14] DUDEK, A.E. (2018). Block bootstrap for periodic characteristics of periodically correlated time series., Journal of Nonparametric Statistics 30(1) 87-124. · Zbl 1391.62165 · doi:10.1080/10485252.2017.1404060
[15] DUDEK, A.E., LESKOW, J., PAPARODITIS, E. and POLITIS, D. (2014). A generalized block bootstrap for seasonal time series., J. Time Ser. Anal. 35 89-114. · Zbl 1301.62086 · doi:10.1002/jtsa.12053
[16] DUDEK, A.E., PAPARODITIS, E. and POLITIS, D. (2016). Generalized Seasonal Tapered Block Bootstrap, Statistics and Probability Letters 115 27-35. · Zbl 1338.62124 · doi:10.1016/j.spl.2016.03.022
[17] EFRON, B. (1979). Bootstrap Methods: Another look at the Jacknife., Ann Statist. 7 1-26. · Zbl 0406.62024 · doi:10.1214/aos/1176344552
[18] FRANKE, J. and HÄRDLE, W. (1992). On bootstrapping kernel spectral estimates., Ann. Statist. 20 121-145. · Zbl 0757.62048
[19] HURVICH, C. M. and ZEGER, S. L. (1987). Frequency domain bootstrap methods for time series. Technical Report 87-115, Graduate School of Business Administration, New York, Univ.
[20] JANAS, D. and DAHLHAUS, R. (1994). A frequency domain bootstrap for time series., In Computationally Intensive Statistical Methods. In Proceedings of the 26th Symposium on the Interface J. Sall and A. Lehman, eds., 423-425. Interface Foundation of North America, Fairfax Station, VA.
[21] KIM, Y.M. and IM, J. (2019). Frequency domain bootstrap for ratio statistics under long-range dependence., Journal of the Korean Statistical Society, 48(4) 547-560. · Zbl 1439.62109 · doi:10.1016/j.jkss.2019.03.001
[22] KIM, Y.M. and NORDMAN, D.J. (2013). A frequency domain bootstrap for Whittle estimation under long-range dependence., Journal of Multivariate Analysis 115C 405-420. · Zbl 1259.62084 · doi:10.1016/j.jmva.2012.10.018
[23] KIRCH, C. and POLITIS, D.N. (2011). TFT-bootstrap: resampling time series in the frequency domain to obtain replicates in the time domain., Ann. of Stat. 39 1427-1470. · Zbl 1220.62107 · doi:10.1214/10-AOS868
[24] KREISS, J.-P. and PAPARODITIS, E. (2003). Autoregressive aided periodogram bootstrap for time series., Ann. Stat. 31 1923-1955. · Zbl 1042.62081 · doi:10.1214/aos/1074290332
[25] KÜNSCH, H. (1989). The jackknife and the bootstrap for general stationary observations., Ann. Statist. 17 1217-1241. · Zbl 0684.62035
[26] Lahiri, S.N. (2003)., Resampling Methods for Dependent Data. Springer, New York. · Zbl 1028.62002
[27] LIU, R. and SINGH, K. (1992). Moving block jackknife and bootstrap capture weak dependence., Exploring the Limits of Bootstrap. Wiley Ser. Probab. Math. Statist. Probab. Math. Statist. Wiley, New York, pp 225-248. · Zbl 0838.62036
[28] MEYER, M., PAPARODITIS, E. and KREISS, J.-P. (2020). Extending the validity of frequency domain bootstrap methods to general stationary processes., Ann. Statist. 48(4) 2404-2427. · Zbl 1458.62207 · doi:10.1214/19-AOS1892
[29] NORDGAARD, A. (1992). Resampling stochastic processes using a bootstrap approach., In Bootstrapping and Related Techniques. Lecture Notes in Econom. Math. Sys. 376. Springer, Berlin.
[30] PAPARODITIS, E. (2000). Spectral density based goodness-of-fit tests for time series models, Scandinavian Journal of Statistics 27 143-176. · Zbl 0940.62084 · doi:10.1111/1467-9469.00184
[31] PAPARODITIS, E. (2002). Frequency domain bootstrap for time series., In Empirical Process Techniques for Dependent Data (H. Dehling et al., eds.) 365-381. Birkhäuser, Boston. · Zbl 1021.62030
[32] PAPARODITIS, E. and POLITIS, D. N. (1999). The local bootstrap for periodogram statistics., J. Time Series Analysis 20 193-222. · Zbl 0938.62051 · doi:10.1111/1467-9892.00133
[33] PAPARODITIS, E. and POLITIS, D. (2001). Tapered block bootstrap., Biometrika 88 1105-1119. · Zbl 0987.62027 · doi:10.1093/biomet/88.4.1105
[34] PATTON, A., POLITIS, D. and WHITE, H. (2009). Correction to “Automatic Block-length Selection for the Dependent Bootstrap”, Econometric Reviews 28 372-375. · Zbl 1400.62193 · doi:10.1080/07474930802459016
[35] POLITIS, D. N. (2001). Resampling time series with seasonal components, in, Frontiers in Data Mining and Bioinformatics: Proceedings of the 33rd Symposium on the Interface of Computing Science and Statistics, Orange County, California, June 13-17, pp. 619-621.
[36] POLITIS, D. N. (2003). Adaptive bandwidth choice., Journal of Nonparametric Statistics 15(4-5) 517-533. · Zbl 1054.62038 · doi:10.1080/10485250310001604659
[37] POLITIS, D.N. and ROMANO, J.P. (1992). A circular block-resampling procedure for stationary data., Exploring the Limits of Bootstrap. Wiley Ser. Probab. Math. Statist. Probab. Math. Statist. Wiley, New York, pp 263-270. · Zbl 0845.62036
[38] POLITIS, D.N and ROMANO, J.P. (1994). The stationary bootstrap., J. Amer. Statist. Assoc. 89 1303-1313. · Zbl 0814.62023 · doi:10.1080/01621459.1994.10476870
[39] POLITIS, D. N. and WHITE, H. (2004). Automatic block-length selection for the dependent bootstrap., Econometric Reviews 23 53-70. · Zbl 1082.62076 · doi:10.1081/ETC-120028836
[40] PONS, O. and TURCKHEIM, E. (1991). Von Mises method, Bootstrap and Hadamard differentiability, Statistics 22 205-214.
[41] PRIESTLEY, M.B. (1981)., Spectral Analysis and Time Series Analysis. Academic Press, London. · Zbl 0537.62075
[42] RIEDER, H. (1994)., Robust Asymptotic Statistics. Springer Verlag N.Y. · Zbl 0927.62050
[43] SHIMIZU, K. (2010)., Bootstrapping Stationary ARMA-GARCH Models. Vieweg+Teubner.
[44] VAN DER VAART, A.W. and WELLNER, J.A.W (1996)., Weak convergence and empirical processes, with applications to statistics, Springer verlag, N.Y. · Zbl 0862.60002
[45] WU, C. · Zbl 0618.62072 · doi:10.1214/aos/1176350142
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