Testing relevant hypotheses in functional time series via self-normalization

Holger Dette, Kevin Kokot, Stanislav Volgushev

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2020 · Volume 82 · Issue 3

Abstract

We develop methodology for testing relevant hypotheses about functional time series in a tuning-free way. Instead of testing for exact equality, e.g. for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of norelevantdeviation. In the two-sample problem this means that anL2-distance between the two mean functions is smaller than a prespecified threshold. For such hypotheses self-normalization, which was introduced in 2010 by Shao, and Shao and Zhang and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self-normalized procedures for testing relevant hypotheses in the one-sample, two-sample and change point problem and investigate their asymptotic properties. Finite sample properties of the tests proposed are illustrated by means of a simulation study and data examples. Our main focus is on functional time series, but extensions to other settings are also briefly discussed.

Keywords

Change point analysis, Cumulative sum, Functional time series, Relevant hypotheses, Self-normalization, Two-sample problems

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