Change-Point Detection and Bootstrap for Hilbert Space Valued Random Fields

TL;DR

This paper develops a new statistical test for detecting epidemic changes in Hilbert space valued random fields, utilizing bootstrap methods and functional limit theorems to improve inference without variance estimation.

Contribution

It introduces a Cramér-von Mises type test for general distribution changes in random fields, and adapts Shao's bootstrap for Hilbert space valued data, supported by theoretical proofs.

Findings

The test is consistent for epidemic change detection.

Bootstrap method provides accurate critical values.

Simulation confirms theoretical results.

Abstract

The problem of testing for the presence of epidemic changes in random fields is investigated. In order to be able to deal with general changes in the marginal distribution, a Cram\'er-von Mises type test is introduced which is based on Hilbert space theory. A functional central limit theorem for $\rho$-mixing Hilbert space valued random fields is proven. In order to avoid the estimation of the long-run variance and obtain critical values, Shao's dependent wild bootstrap method is adapted to this context. For this, a joint functional central limit theorem for the original and the bootstrap sample is shown. Finally, the theoretic results are supplemented by a short simulation study.