In the common time series model $X_{i,n} = mu (i/n) + varepsilon_{i,n}$ with non-stationary errors we consider the problem of detecting a significant deviation of the mean function $mu$ from a benchmark $g (mu )$ (such as the initial value $mu (0)$ or the average trend $int_{0}^{1} mu (t) dt$). 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 $ (mu (i/n))_{i =1,ldots ,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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