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April 2020 Functional data analysis in the Banach space of continuous functions
Holger Dette, Kevin Kokot, Alexander Aue
Ann. Statist. 48(2): 1168-1192 (April 2020). DOI: 10.1214/19-AOS1842

Abstract

Functional data analysis is typically conducted within the $L^{2}$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same time, there have recently been a number of publications that sidestep dimension reduction steps and focus on a fully functional $L^{2}$-methodology. This paper goes one step further and develops data analysis methodology for functional time series in the space of all continuous functions. The work is motivated by the fact that objects with rather different shapes may still have a small $L^{2}$-distance and are therefore identified as similar when using a $L^{2}$-metric. However, in applications it is often desirable to use metrics reflecting the visualization of the curves in the statistical analysis. The methodological contributions are focused on developing two-sample and change-point tests as well as confidence bands, as these procedures appear to be conducive to the proposed setting. Particular interest is put on relevant differences; that is, on not trying to test for exact equality, but rather for prespecified deviations under the null hypothesis.

The procedures are justified through large-sample theory. To ensure practicability, nonstandard bootstrap procedures are developed and investigated addressing particular features that arise in the problem of testing relevant hypotheses. The finite sample properties are explored through a simulation study and an application to annual temperature profiles.

Citation

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Holger Dette. Kevin Kokot. Alexander Aue. "Functional data analysis in the Banach space of continuous functions." Ann. Statist. 48 (2) 1168 - 1192, April 2020. https://doi.org/10.1214/19-AOS1842

Information

Received: 1 October 2017; Revised: 1 February 2019; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241585
MathSciNet: MR4102692
Digital Object Identifier: 10.1214/19-AOS1842

Subjects:
Primary: 62G10, 62G15, 62M10

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • April 2020
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