Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators
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Date
2020
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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 Xi = μ(i/n) +εi 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 1
0 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
ˆ d∞n, for the maximum deviation d∞ = supt∈[x0,x1] |μ(t) − g(μ)|. We derive the
limiting distribution of an appropriately standardized version of ˆ 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.
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Keywords
relevant change point analysis, gradual changes, maximum deviation, local-linear estimator, Gumbel distribution, Gaussian approximation
Subjects based on RSWK
Change-point-Problem, Maximale Abweichung, Schätzfunktion, Gauß-Approximation