Testing relevant hypotheses in functional time series via self-normalization

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.

Two-sample tests for relevant differences in the eigenfunctions of covariance operators

“…As a remedy, self-normalization is a tuning parameter-free method that achieves standardization, typically through recursive estimates. The advantages of such an approach for testing relevant hypotheses of parameters of functional time series were recently recognized in Dette et al (2018). In this paper, we develop a concept of self-normalization for the problem of testing for relevant differences between the eigenfunctions of two covariance operators in functional data.…”

Detecting structural breaks in eigensystems of functional time series

“…We now state a first result concerning the convergence rate of the change point estimator defined in (2.6). The proof follows by similar arguments as given in the proof of Proposition 3.1 in Dette et al (2018), which are omitted for the sake of brevity.…”

“…This concept has been introduced for change point detection in a seminal paper by Shao and Zhang (2010) and since then been used by many authors. While most of this literature concentrates on classical change point problems, Dette et al (2018) introduced a novel type of self-normalization for relevant hypotheses and used it to define a self-normalized test for a relevant change in the mean of a time series. In the following we will further develop this concept to detect relevant changes in the spectrum.…”

“…Relevant hypotheses have been considered in statistics to different degrees since the mid 20th century (see Hodges and Lehmann (1954)), and have been investigated intensively in biostatistics, where tests for "bioequivalence" of certain drugs have nowadays become standard (see for example Wellek (2010)). In the context of change point analysis for functional data relevant hypotheses have recently been considered by Dette et al (2019) for Banach-space valued random variables and by Dette et al (2018) in Hilbert spaces. The first named paper concentrates on inference regarding the mean functions while Dette et al (2018) developed tests for a relevant structural break in the mean function and in the covariance operator.…”

“…In the context of change point analysis for functional data relevant hypotheses have recently been considered by Dette et al (2019) for Banach-space valued random variables and by Dette et al (2018) in Hilbert spaces. The first named paper concentrates on inference regarding the mean functions while Dette et al (2018) developed tests for a relevant structural break in the mean function and in the covariance operator. The detection of structural breaks in the eigenvalues and eigenfunctions is a substantially more difficult problem due to their implicit definition and statistical tests have mainly been developed for the two sample case (see Zhang and Shao (2015), who consider classical hypotheses and Aue et al (2019), who discuss relevant hypotheses).…”