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

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 Xi = μ (i/n) + varepsiloni 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 int01 μ (t) dt) is smaller than a given threshold on a given interval [x0,x1] subseteq [0,1]. A test for this type of hypotheses is developed using an appropriate estimator, say hat 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 hat 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.