Change point analysis in non-stationary processes - a mass excess approach

This paper considers the problem of testing if a sequence of means $({\mu }_t{)}_{t=1,\dots ,n}$ of a non-stationary time series $(X_t)_{t =1,\ldots ,n }$isstableinthesensethatthedifferenceofthemeans$\mu_1$and$\mu_t$betweentheinitialtime$t=1$ and any other time is smaller than a given level, that is $\mathrm{\mid }{\mu }_1-{\mu }_t\mathrm{\mid }\le c$ for all $t=1,\dots ,n$. A test for hypotheses of this type is developed using a biascorrected monotone rearranged local linear estimator and asymptotic normality of the corresponding test statistic is established. As the asymptotic variance depends on the location and order of the critical roots of the equation $| \mu_1 - \mu_t | = c$ a new bootstrap procedure is proposed to obtain critical values and its consistency is established. As a consequence we are able to quantitatively describe relevant deviations of a non-stationary sequence from its initial value. The results are illustrated by means of a simulation study and by analyzing data examples.