On joint testing of changes in conditional mean and variance functions of stationary and ergodic time series

Detecting changes in the behavior of time series is an important issue in statistical analysis. This paper introduces a framework for simultaneously testing changes in the conditional mean and variance functions of stationary and ergodic time series. These tests are based on functionals of weighted cumulative sum (CUSUM) processes. In a semi-parametric framework, we establish the weak convergence of this family of processes under the null hypothesis of no change-point, assuming stationarity and allowing the errors to exhibit dependence while forming a martingale difference sequence. Test statistics, derived as functionals of these processes, are shown to have distribution-free limits. This approach unifies many existing change-point tests. Monte Carlo simulations are conducted to illustrate the validity of our approach and to compare the proposed tests with existing methods from the change-point literature. Additionally, the testing procedures are applied to two real data examples.