A distribution free test for changes in the trend function of locally stationary processes
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In the common time series model X_i,n = μ (i/n) + ε_i,n with non-stationary errors we consider the problem of detecting a significant deviation of the mean function μ from a benchmark g (μ ) (such as the initial value μ (0) or the average trend ∫_0^1μ (t) dt). The problem is motivated by a more realistic modelling of change point analysis, where one is interested in identifying relevant deviations in a smoothly varying sequence of means (μ (i/n))_i =1,... ,n and cannot assume that the sequence is piecewise constant. A test for this type of hypotheses is developed using an appropriate estimator for the integrated squared deviation of the mean function and the threshold. By a new concept of self-normalization adapted to non-stationary processes an asymptotically pivotal test for the hypothesis of a relevant deviation is constructed. The results are illustrated by means of a simulation study and a data example.
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