Estimation of change-points in a nonparametric regression function through kernel density estimation

Abstract

In a stable system, the process under study usually has a constant mean. An action applied to the system may cause an effect instantly or gradually on the mean of the process. After a period of time, the effect of the action may become stable and it may drive the mean of the process to another constant instantly or gradually. In this case, the mean of the process before it is affected by the action and that after it becomes constant again are estimated. The times at which the mean of the process is affected by the action and when it becomes constant again are also estimated. The estimators for these quantities are analyzed by central limit theorems (CLT) and strong convergence rates (SCR). The CLT for the estimators of the above two means are the same as those for the sample means constructed in the case that the above two times are known in advance. The smoothness of the change in the mean function has no effect on both the CLT and SCR for the estimators of the two means. But it does have an effect on the estimators for the above two times. This effect is quantified precisely through the order of the SCR of the estimators. Simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes.

Keywords:

• change–point
• nonparametric regression
• kernel density estimator
• asymptotic normality
• strong convergence rate
• cross–validation