Unified theory of testing relevant hypothesis in functional time series

TL;DR

This paper introduces a comprehensive, nuisance-parameter-free testing framework for relevant hypotheses in functional time series, applicable to various problems including change points, with robust theoretical foundations and practical demonstrations.

Contribution

It develops a unified, self-normalized testing approach for functional time series that handles complex scenarios without estimating nuisance parameters, addressing non-tightness issues via Gaussian approximation.

Findings

Effective in sparse and dense sampling regimes

Supports multiple change point detection with consistent estimation

Validated through extensive simulations and real data applications

Abstract

In this paper, we present a general framework for testing relevant hypotheses in functional time series. Our unified approach covers one-sample, two-sample, and change point problems under contaminated observations with arbitrary sampling schemes. By employing B-spline estimators and the self-normalization technique, we propose nuisance-parameter-free testing procedures, obviating the need for additional procedures such as estimating long-run covariance or measurement-error variance functions. A key challenge arises from related nonparametric statistics may not be tight, complicating the joint weak convergence for the test statistics and self-normalizers, particularly in sparse scenarios. To address this, we leverage a Gaussian approximation in a diverging-dimension regime to derive a pivotal approximate distribution. Then, we develop consistent decision rules, provide sufficient conditions ensuring non-degeneracy, and establish phase transition boundaries from sparse to dense. We also examine the multiple change point scenario and extend the theory when one obtains consistent estimates of the change points. The choice of self-normalizers is further discussed, including the recently developed range-adjusted self-normalizer. Extensive numerical experiments support the proposed theory, and we illustrate our methodologies using the AU.SHF implied volatility and traffic volume datasets.