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ARE DEVIATIONS in A GRADUALLY VARYING MEAN RELEVANT? A TESTING APPROACH BASED on SUP-NORM ESTIMATORS

Bücher, A. and Dette, H. and Heinrichs, F.

ANNALS OF STATISTICS
Volume: 49 Pages: 3583-3617
DOI: 10.1214/21-AOS2098
Published: 2021

Abstract
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 Xi = μ(i/n) + εi with centered errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function μ from a functional g(μ) (such as the value μ(0) or the integral 01 μ(t)dt) is smaller than a given threshold on a given interval [x0, x1] ⊆ [0, 1]. A test for this type of hypotheses is developed using an appropriate estimator, say d∞,n, for the maximum deviation d∞ = supt∈[x0,x1] |μ(t) − g(μ)|. We derive the limiting distribution of an appropriately standardized version of d∞,n, where the standardization depends on the Lebesgue measure of the set of extremal points of the function μ(·) − g(μ). 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. © Institute of Mathematical Statistics, 2021

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