Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators
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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 ∫_0^1μ (t) dt) 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.
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