Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators

2020/02/14 by Axel Bücher, Holger Dette, Bücher, Axel +3 · 3 citations
Environmental Science · Mathematics · #62G08 #62M10 #Environmental Impact and Sustainability #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Point processes and geometric inequalities #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2002.06143

openalex publication_date 2020/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly non-stationary 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 non-relevant (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 = μ(i/n) + ε_i with possibly non-stationary 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_∞= sup_t ∈ [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.

Cited by

• Testing relevant difference in high-dimensional linear regression with applications to detect transferability
• Detection of mean changes in partially observed functional data
• Self-Normalization for CUSUM-based Change Detection in Locally Stationary Time Series

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