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 nonstationary time series and (2) identifying regions where exhibits piecewise constant behavior. In many applications however, it is more reasonable to assume that gradually smooth way. Those gradual may either be nonrelevant (i.e., small), or relevant for specific problem hand, present paper presents statistical methodology detect latter. More precisely, we consider common nonparametric regression model Xi=μ(i/n)+εi with centered errors propose test null hypothesis maximum absolute deviation function μ from functional g(μ) (such as value μ(0) integral ∫01μ(t)dt) smaller than given threshold on interval [x0,x1]⊆[0,1]. A this type hypotheses developed using an appropriate estimator, say dˆ∞,n, d∞=supt∈[x0,x1]|μ(t)−g(μ)|. We derive limiting distribution appropriately standardized version standardization depends Lebesgue measure set extremal points μ(·)−g(μ). refined procedure based estimate its consistency proved. The results are illustrated by means simulation study data example.