A surveillance procedure for random walks based on local linear estimation

Abstract

We study the problem of detecting a change in the trend of time series whose stochastic part behaves as a random walk. Interesting applications in finance and engineering come to mind. Local linear estimation provides a well established approach to estimating level and derivative of an underlying trend function, provided the time instants where observations are available become dense asymptotically. Here we study the local linear estimation principle for the classic time series setting where the distance between time points is fixed. It turns out that local linear estimation is applicable to the detection problem, and we identify the underlying (asymptotic) parameters. Assuming that observations arrive sequentially, we propose surveillance procedures and establish the relevant asymptotic theory, particularly, an invariance principle for the sequential empirical local linear process. A simulation study illustrates the remarkable accuracy of the approximations obtained by our asymptotics even in small samples as well as the excellent detection performance of the proposed surveillance procedure.

Keywords:

• change point
• non-standard asymptotics
• invariance principle
• nonparametric regression
• sequential analysis
• weak dependence

AMS Subject Classification :

• 62L10
• 62E20
• 62G08
• 60G50

Acknowledgements

The author thanks two anonymous referees for constructive remarks which improved the presentation of the results.