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CUSUM chart for detecting range shifts when monotonicity of likelihood ratio is invalid

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  • Guanfu Liu
  • Xiaolong Pu
  • Lei Wang
  • Dongdong Xiang

Abstract

It is often encountered in the literature that the log-likelihood ratios (LLR) of some distributions (e.g. the student t distribution) are not monotonic. Existing charts for monitoring such processes may suffer from the fact that the average run length (ARL) curve is a discontinuous function of control limit. It implies that some pre-specified in-control (IC) ARLs of these charts may not be reached. To guarantee the false alarm rate of a control chart lower than the nominal level, a larger IC ARL is usually suggested in the literature. However, the large IC ARL may weaken the performance of a control chart when the process is out-of-control (OC), compared with a just right IC ARL. To overcome it, we adjust the LLR to be a monotonic one in this paper. Based on it, a multiple CUSUM chart is developed to detect range shifts in IC distribution. Theoretical result in this paper ensures the continuity of its ARL curve. Numerical results show our proposed chart performs well under the range shifts, especially under the large shifts. In the end, a real data example is utilized to illustrate our proposed chart.

Suggested Citation

  • Guanfu Liu & Xiaolong Pu & Lei Wang & Dongdong Xiang, 2015. "CUSUM chart for detecting range shifts when monotonicity of likelihood ratio is invalid," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(8), pages 1635-1644, August.
    • Handle: RePEc:taf:japsta:v:42:y:2015:i:8:p:1635-1644
    DOI: 10.1080/02664763.2015.1004625
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    References listed on IDEAS

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    1. Han, Dong & Tsung, Fugee, 2006. "A Reference-Free Cuscore Chart for Dynamic Mean Change Detection and a Unified Framework for Charting Performance Comparison," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 368-386, March.
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    3. Luo, Yunzhao & Li, Zhonghua & Wang, Zhaojun, 2009. "Adaptive CUSUM control chart with variable sampling intervals," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2693-2701, May.
    4. MacEachern, Steven N. & Rao, Youlan & Wu, Chunjie, 2007. "A Robust-Likelihood Cumulative Sum Chart," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1440-1447, December.
    5. P. J. Harrison, 1999. "Statistical process control and model monitoring," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(2), pages 273-292.
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