  Most of the literature on change-point analysis by means of hypothesis
testing considers hypotheses of the form H0 : \theta_1 = \theta_2 vs. H1 :
\theta_1 != \theta_2, where \theta_1 and \theta_2 denote parameters of the
process before and after a change point. This paper takes a different
perspective and investigates the null hypotheses of no relevant changes, i.e.
H0 : ||\theta_1 - \theta_2|| ? \leq \Delta?, where || \cdot || is an
appropriate norm. This formulation of the testing problem is motivated by the
fact that in many applications a modification of the statistical analysis might
not be necessary, if the difference between the parameters before and after the
change-point is small. A general approach to problems of this type is developed
which is based on the CUSUM principle. For the asymptotic analysis weak
convergence of the sequential empirical process has to be established under the
alternative of non-stationarity, and it is shown that the resulting test
statistic is asymptotically normal distributed. Several applications of the
methodology are given including tests for relevant changes in the mean,
variance, parameter in a linear regression model and distribution function
among others. The finite sample properties of the new tests are investigated by
means of a simulation study and illustrated by analyzing a data example from
economics.
