Testing relevant hypotheses in functional time series via self-normalization

TL;DR

This paper introduces a new self-normalization methodology for testing relevant hypotheses in functional time series, allowing for tuning-free, robust inference about mean functions and change points.

Contribution

It develops novel self-normalized tests for relevant hypotheses in functional time series, extending existing methods to handle deviations beyond exact equality.

Findings

Proposed tests have good finite sample performance.

Methodology effectively detects relevant deviations in mean functions.

Extensions to other settings are briefly discussed.

Abstract

In this paper we develop methodology for testing relevant hypotheses about functional time series in a tuning-free way. Instead of testing for exact equality, for example for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of no relevant deviation. In the two sample problem this means that an $L^2$-distance between the two mean functions is smaller than a pre-specified threshold. For such hypotheses self-normalization, which was introduced by Shao (2010) and Shao and Zhang (2010) 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 proposed tests 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.