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.


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


fregion; fda (R)
Full Text: DOI arXiv Euclid


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