ARE DEVIATIONS IN A GRADUALLY VARYING MEAN RELEVANT? A TESTING APPROACH BASED ON SUP-NORM ESTIMATORS

• 成果类型: Article
• 署名作者: Buecher, Axel; Dette, Holger; Heinrichs, Florian
• 署名单位: Heinrich Heine University Dusseldorf; Ruhr University Bochum
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2098
• 发表日期: 2021
• 页码: 3583-3617
• 关键词: multiple change-points; confidence bands; time-series; structural-change; regression; inference; TRENDS

摘要

Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly nonstationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be nonrelevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model X-i = mu(i/n) + epsilon(i) with centered errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function mu from a functional g(mu) (such as the value mu(0) or the integral integral(1)(0)mu(t)dt) is smaller than a given threshold on a given interval [x(0), x(1)] subset of [0, 1]. A test for this type of hypotheses is developed using an appropriate estimator, say (d) over cap (infinity,n), for the maximum deviation d(infinity) = sup(t is an element of[x0, x1]) vertical bar mu(tr) - g(mu)vertical bar. We derive the limiting distribution of an appropriately standardized version of (d) over cap (infinity,n), where the standardization depends on the Lebesgue measure of the set of extremal points of the function mu(center dot) - g(mu). A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example.