# Consistent nonparametric change point detection combining CUSUM and marked empirical processes

## TL;DR

This paper introduces a new nonparametric change point detection method for weakly dependent time series, combining CUSUM and empirical processes, with proven convergence and bootstrap applicability.

## Contribution

It develops a novel change point test based on a modified CUSUM and empirical process, with proven weak convergence and a simple limiting distribution.

## Key findings

- The test has a simple limiting distribution under the null hypothesis.
- The procedure is consistent and applicable with bootstrap methods.
- It effectively detects changes in the conditional mean function.

## 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 Cram\'er-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.