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Functional data analysis in the Banach space of continuous functions. (English) Zbl 1456.62085

The paper develops data analysis methodology for functional time series in the space of all continuous functions because, according to the authors, all functions utilized for practical purpose are continuous. The paper consists of five sections. In the second section, authors of the paper provide some basic facts about Central Limit Theorem and Invariance Principle for \(C(T)\)-valued random variables, where \(C(T)\) is the set of continuous functions from \(T\) into real line \(\mathbb{R}\).
The third section deals with the two sample problem on the space \(C([0,1])\). Let \(\{X_1,\ldots,X_m\}\) and \(\{Y_1,\ldots,Y_n\}\) be two independent samples of \(C([0,1])\)-valued random variables. Under suitable assumptions expectation functions \(\mu_1=\mathbb{E}X_1\) and \(\mu_2=\mathbb{E}Y_1\) exist together with the covariance kernels. The authors of the paper consider properties of the maximal deviation between two mean curves \[d_\infty=\|\mu_1-\mu_2\|=\sup_{t\in[0,1]}|\mu_1(t)-\mu_2(t)|\] and provide procedure for testing the hypotheses of relevant difference: \[ H_0: d_\infty\leqslant \Delta\ \ {\text{versus}}\ \ H_1: d_\infty>\Delta, \] where \(\Delta\geqslant 0\) is a constant determined by the user of test.
In the fourth section of the paper, the change point problem is considered. The new results are presented for testing of a change-point for triangular arrays of \(C([0,1])\)-valued random variables satisfying suitable requirements with respect to metric \(\rho(s,t)=|s-t|^\theta\), \(\theta\in(0,1]\).
The simulation study of the derived procedures is described in the last section of the paper. The detailed proofs of the new results and the detailed simulation study investigating the finite sample properties of the new methodology are given in the supplementary materials doi:10.1214/19-AOS1842SUPPA and doi:10.1214/19-AOS1842SUPPB.

MSC:

62G10 Nonparametric hypothesis testing
62G15 Nonparametric tolerance and confidence regions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62R10 Functional data analysis
60F05 Central limit and other weak theorems

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Full Text: DOI arXiv Euclid

References:

[1] Aston, J. A. D. and Kirch, C. (2012). Detecting and estimating changes in dependent functional data. J. Multivariate Anal. 109 204-220. · Zbl 1241.62121 · doi:10.1016/j.jmva.2012.03.006
[2] Aston, J. A. D. and Kirch, C. (2012). Evaluating stationarity via change-point alternatives with applications to fMRI data. Ann. Appl. Stat. 6 1906-1948. · Zbl 1257.62072 · doi:10.1214/12-AOAS565
[3] Aue, A., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Estimation of a change-point in the mean function of functional data. J. Multivariate Anal. 100 2254-2269. · Zbl 1176.62025 · doi:10.1016/j.jmva.2009.04.001
[4] Aue, A. and Horváth, L. (2013). Structural breaks in time series. J. Time Series Anal. 34 1-16. · Zbl 1274.62553
[5] Aue, A., Norinho, D. D. and Hörmann, S. (2015). On the prediction of stationary functional time series. J. Amer. Statist. Assoc. 110 378-392. · Zbl 1373.62462 · doi:10.1080/01621459.2014.909317
[6] Aue, A., Rice, G. and Sönmez, O. (2018). Detecting and dating structural breaks in functional data without dimension reduction. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 509-529. · Zbl 1398.62152 · doi:10.1111/rssb.12257
[7] Berkes, I., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Detecting changes in the mean of functional observations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 927-946. · Zbl 1411.62153 · doi:10.1111/j.1467-9868.2009.00713.x
[8] Berkes, I., Horváth, L. and Rice, G. (2013). Weak invariance principles for sums of dependent random functions. Stochastic Process. Appl. 123 385-403. · Zbl 1269.60040 · doi:10.1016/j.spa.2012.10.003
[9] Berkson, J. (1938). Some difficulties of interpretation encountered in the application of the chi-square test. J. Amer. Statist. Assoc. 33 526-536. · Zbl 0019.17701 · doi:10.1080/01621459.1938.10502329
[10] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[11] Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. Springer, New York. · Zbl 0962.60004
[12] Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107-144. · Zbl 1189.60077 · doi:10.1214/154957805100000104
[13] Bucchia, B. and Wendler, M. (2017). Change-point detection and bootstrap for Hilbert space valued random fields. J. Multivariate Anal. 155 344-368. · Zbl 1356.62166 · doi:10.1016/j.jmva.2017.01.007
[14] Bücher, A. and Kojadinovic, I. (2016). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing. Bernoulli 22 927-968. · Zbl 1388.62123 · doi:10.3150/14-BEJ682
[15] Cao, G. (2014). Simultaneous confidence bands for derivatives of dependent functional data. Electron. J. Stat. 8 2639-2663. · Zbl 1309.62074 · doi:10.1214/14-EJS967
[16] Cao, G., Yang, L. and Todem, D. (2012). Simultaneous inference for the mean function based on dense functional data. J. Nonparametr. Stat. 24 359-377. · Zbl 1241.62119 · doi:10.1080/10485252.2011.638071
[17] Choi, H. and Reimherr, M. (2018). A geometric approach to confidence regions and bands for functional parameters. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 239-260. · Zbl 1381.62143 · doi:10.1111/rssb.12239
[18] Degras, D. A. (2011). Simultaneous confidence bands for nonparametric regression with functional data. Statist. Sinica 21 1735-1765. · Zbl 1225.62052 · doi:10.5705/ss.2009.207
[19] Dehling, H. (1983). Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrsch. Verw. Gebiete 63 393-432. · Zbl 0496.60004 · doi:10.1007/BF00542537
[20] Dehling, H. and Philipp, W. (1982). Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. Probab. 10 689-701. · Zbl 0487.60006 · doi:10.1214/aop/1176993777
[21] Dette, H., Kokot, K. and Aue, A. (2020). Supplement to “Functional data analysis in the Banach space of continuous functions.” https://doi.org/10.1214/19-AOS1842SUPPA, https://doi.org/10.1214/19-AOS1842SUPPB.
[22] Dette, H. and Wied, D. (2016). Detecting relevant changes in time series models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 371-394. · Zbl 1414.62360 · doi:10.1111/rssb.12121
[23] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. Springer, New York. · Zbl 1119.62046
[24] Fremdt, S., Horváth, L., Kokoszka, P. and Steinebach, J. G. (2014). Functional data analysis with increasing number of projections. J. Multivariate Anal. 124 313-332. · Zbl 1359.62197 · doi:10.1016/j.jmva.2013.11.009
[25] Garling, D. J. H. (1976). Functional central limit theorems in Banach spaces. Ann. Probab. 4 600-611. · Zbl 0343.60014 · doi:10.1214/aop/1176996030
[26] Hall, P. and Van Keilegom, I. (2007). Two-sample tests in functional data analysis starting from discrete data. Statist. Sinica 17 1511-1531. · Zbl 1136.62035
[27] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845-1884. · Zbl 1189.62141 · doi:10.1214/09-AOS768
[28] Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. Springer, New York. · Zbl 1279.62017
[29] Horváth, L., Kokoszka, P. and Reeder, R. (2013). Estimation of the mean of functional time series and a two-sample problem. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 103-122. · Zbl 07555440
[30] Horváth, L., Kokoszka, P. and Rice, G. (2014). Testing stationarity of functional time series. J. Econometrics 179 66-82. · Zbl 1293.62186 · doi:10.1016/j.jeconom.2013.11.002
[31] Hughes, G. L., Subba Rao, S. and Subba Rao, T. (2007). Statistical analysis and time-series models for minimum/maximum temperatures in the Antarctic Peninsula. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 241-259. · Zbl 1128.62122 · doi:10.1098/rspa.2006.1766
[32] Janson, S. and Kaijser, S. (2015). Higher moments of Banach space valued random variables. Mem. Amer. Math. Soc. 238 vii+110. · Zbl 1336.60007
[33] Kuelbs, J. (1973). The invariance principle for Banach space valued random variables. J. Multivariate Anal. 3 161-172. · Zbl 0258.60009 · doi:10.1016/0047-259X(73)90020-1
[34] Kuelbs, J. and Philipp, W. (1980). Almost sure invariance principles for partial sums of mixing \(B\)-valued random variables. Ann. Probab. 8 1003-1036. · Zbl 0451.60008 · doi:10.1214/aop/1176994565
[35] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Springer, Berlin. · Zbl 0748.60004
[36] Paparoditis, E. and Politis, D. N. (2001). Tapered block bootstrap. Biometrika 88 1105-1119. · Zbl 0987.62027 · doi:10.1093/biomet/88.4.1105
[37] 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
[38] Račkauskas, A. and Suquet, C. (2004). Hölder norm test statistics for epidemic change. J. Statist. Plann. Inference 126 495-520. · Zbl 1084.62083 · doi:10.1016/j.jspi.2003.09.004
[39] Račkauskas, A. and Suquet, C. (2006). Testing epidemic changes of infinite dimensional parameters. Stat. Inference Stoch. Process. 9 111-134. · Zbl 1110.62060 · doi:10.1007/s11203-005-0728-5
[40] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. Springer, New York. · Zbl 1079.62006
[41] Samur, J. D. (1984). Convergence of sums of mixing triangular arrays of random vectors with stationary rows. Ann. Probab. 12 390-426. · Zbl 0542.60012 · doi:10.1214/aop/1176993297
[42] Samur, J. D. (1987). On the invariance principle for stationary \(\phi \)-mixing triangular arrays with infinitely divisible limits. Probab. Theory Related Fields 75 245-259. · Zbl 0594.60039 · doi:10.1007/BF00354036
[43] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, New York. · Zbl 0862.60002
[44] Wellek, S. (2010). Testing Statistical Hypotheses of Equivalence and Noninferiority, 2nd ed. CRC Press, Boca Raton, FL. · Zbl 1219.62002
[45] Zheng, S. · Zbl 1367.62151 · doi:10.1080/01621459.2013.866899
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