Title: Consistent nonparametric change point detection combining CUSUM and marked empirical processes,
Abstract: A weakly dependent time series regression model with multivariate covariates
and univariate observations is considered, for which we develop a procedure to
detect whether the nonparametric conditional mean function is stable in time
against change point alternatives. Our proposal is based on a modified CUSUM
type test procedure, which uses a sequential marked empirical process of
residuals. We show weak convergence of the considered process to a centered
Gaussian process under the null hypothesis of no change in the mean function
and a stationarity assumption. This requires some sophisticated arguments for
sequential empirical processes of weakly dependent variables. As a consequence
we obtain convergence of Kolmogorov-Smirnov and Cramér-von Mises type test
statistics. The proposed procedure acquires a very simple limiting distribution
and nice consistency properties, features from which related tests are lacking.
We moreover suggest a bootstrap version of the procedure and discuss its
applicability in the case of unstable variances.