Two-Sample and Change-Point Inference for Non-Euclidean Valued Time Series

TL;DR

This paper develops new statistical inference methods for non-Euclidean time series data, enabling two-sample testing and change-point detection under dependence, with theoretical guarantees and practical applications.

Contribution

It introduces a self-normalization approach for non-Euclidean time series, deriving new limiting distributions and consistency results, and extends to multiple change-point estimation.

Findings

Proposed tests outperform existing methods in simulations.

Effective change-point detection in mortality and cryptocurrency data.

Theoretical results hold under weaker regularity conditions.

Abstract

Data objects taking value in a general metric space have become increasingly common in modern data analysis. In this paper, we study two important statistical inference problems, namely, two-sample testing and change-point detection, for such non-Euclidean data under temporal dependence. Typical examples of non-Euclidean valued time series include yearly mortality distributions, time-varying networks, and covariance matrix time series. To accommodate unknown temporal dependence, we advance the self-normalization (SN) technique (Shao, 2010) to the inference of non-Euclidean time series, which is substantially different from the existing SN-based inference for functional time series that reside in Hilbert space (Zhang et al., 2011). Theoretically, we propose new regularity conditions that could be easier to check than those in the recent literature, and derive the limiting distributions of the proposed test statistics under both null and local alternatives. For change-point detection problem, we also derive the consistency for the change-point location estimator, and combine our proposed change-point test with wild binary segmentation to perform multiple change-point estimation. Numerical simulations demonstrate the effectiveness and robustness of our proposed tests compared with existing methods in the literature. Finally, we apply our tests to two-sample inference in mortality data and change-point detection in cryptocurrency data.