Journal Article

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

Heinrichs, Florian, Dette, Holger, Bücher, Axel

2021

Overview

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) + ε i 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 ∫ 0 1 μ ( t ) d t ) is smaller than a given threshold on a given interval [x 0, x 1] ⊆ [0, 1]. A test for this type of hypotheses is developed using an appropriate estimator, say d̂ ∞, n , for the maximum deviation d ∞ = sup t ∈ [ x 0 , x 1 ] | μ ( t ) − g ( μ ) | . We derive the limiting distribution of an appropriately standardized version of d̂ ∞, 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.

• Publisher: Institute of Mathematical Statistics
• Subject: Deviation / Diffraction / Estimating techniques / Mathematical functions / Null hypothesis / Regression analysis / Regression models / Standardization / Statistical analysis / Time series