On detecting jumps in time series: nonparametric setting

Abstract

Motivated by applications in statistical quality control and signal analysis, we propose a sequential detection procedure which is designed to detect structural changes, in particular jumps, immediately. This is achieved by modifying a median filter by appropriate kernel-based jump-preserving weights (shrinking) and a clipping mechanism. We aim at both robustness and immediate detection of jumps. Whereas the median approach ensures robust smooths when there are no jumps, the modification ensure immediate reaction to jumps. For general clipping location estimators, we show that the procedure can detect jumps of certain heights with no delay, even when applied to Banach space-valued data. For shrinking medians, we provide an asymptotic upper bound for the normed delay. The finite sample properties are studied by simulations which show that our proposal outperforms classical procedures in certain respects.

Keywords:

• Edge detection
• Nonparametric estimation
• Quality control
• Statistical process control

Acknowledgements

The authors thank an anonymous referee for helpful comments. Part of the article was written during a stay of A. Steland at Technical University of Wrolaw, Poland, which was supported by the Deutsche Forschungsgemeinschaft (DFG, grant Ste 1034/2-1.) The financial support of DFGs SFB 475 ‘Reduction of complexity in multivariate data structures’ is also gratefully acknowledged. E. Rafajlowicz expresses thanks to the Polish Council for Scientific Research for a grant ranging from 2002 to 2005.