🤖 AI Summary
This paper addresses three related hypothesis testing problems in functional time series: one-sample testing, two-sample comparison, and multiple change-point detection. We propose a unified, robust nonparametric testing framework applicable to arbitrary sampling designs—from sparse to dense—and to contaminated observations with measurement error, without requiring estimation of nuisance parameters such as long-run covariance or error variance. Leveraging B-spline-based functional estimation and self-normalization, combined with high-dimensional Gaussian approximation theory, our approach overcomes the challenges of insufficient tightness and joint weak convergence of nonparametric test statistics, and rigorously characterizes the sparse–dense phase transition boundary. The theoretical guarantees hold under a diverging-dimension regime. Extensive numerical experiments and real-data applications—including AU.SHF implied volatility and urban traffic flow—demonstrate excellent finite-sample performance and practical utility.
📝 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.