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

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 ${\mathit{X}}_{\mathit{i}}=\mathit{\mu }(\mathit{i}/\mathit{n})+{\mathit{\epsilon }}_{\mathit{i}}$ with centered errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function μ from a functional $\mathit{g}(\mathit{\mu })$ (such as the value $\mathit{\mu }(0)$ or the integral ${\int }_0^1\mathit{\mu }(\mathit{t})\mathrm{d}\mathit{t}$) is smaller than a given threshold on a given interval $[{\mathit{x}}_0,{\mathit{x}}_1]\subseteq [0,1]$. A test for this type of hypotheses is developed using an appropriate estimator, say ${\hat{\mathit{d}}}_{\infty ,\mathit{n}}$, for the maximum deviation ${\mathit{d}}_{\infty }={sup}_{\mathit{t}\in [{\mathit{x}}_0,{\mathit{x}}_1]}|\mathit{\mu }(\mathit{t})-\mathit{g}(\mathit{\mu })|$. We derive the limiting distribution of an appropriately standardized version of ${\hat{\mathit{d}}}_{\infty ,\mathit{n}}$, where the standardization depends on the Lebesgue measure of the set of extremal points of the function $\mathit{\mu }(·)-\mathit{g}(\mathit{\mu })$. 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.