A combined Shewhart-CUSUM chart with switching limit

Abstract

The common Shewhart-cumulative sum (CUSUM) chart deploys an additional Shewhart limit to expand a single CUSUM chart by triggering quick alarms for large changes in the parameter of interest. We utilize this supplementary limit to initiate the CUSUM accumulation, that is, switching between an accumulation phase and a silent phase. The new switching limit’s value resides between the reference value of the CUSUM chart and the usual Shewhart limit. Thus, for the case that the CUSUM statistic is equal to zero, a further observation has to be more substantial than this new limit to engage the summing process. We demonstrate the setup and analyze the new combination for independent Poisson distributed data as well as for a more involved time series model with Poisson marginals, namely, the Poisson first-order integer-valued autoregressive model. Moreover, we also consider a real data set from semiconductor industry with apparently overdispersed counts as well as the application to Gaussian variables data. It turns out that the new chart features patterns between a pure CUSUM and a stand-alone Shewhart chart. Hence, it is a solid alternative to both single charts and the ordinary Shewhart-CUSUM chart.

Keywords:

• CUSUM charts
• run length performance
• Shewhart charts
• statistical process control
• switching limit

About the authors

Sebastian Ottenstreuer is a research assistant in the Department of Mathematics and Statistics at the Helmut Schmidt University in Hamburg, Germany. He received his B.Sc. (2013) and M.Sc. (2016) degrees in Economathematics from the University of Würzburg, Germany.

Christian Weiß is a Professor at the Department of Mathematics and Statistics at the Helmut Schmidt University in Hamburg, Germany. His research areas include time series analysis, statistical quality control, and computational statistics.

Sven Knoth is a Professor at the Department of Mathematics and Statistics at the Helmut Schmidt University in Hamburg, Germany. His research areas include statistical process control, computational statistics, and engineering statistics.

Acknowledgments

The authors thank the referee for carefully reading the article and for the comments, which greatly improved the article.

Notes

1 Actually, this makes the $\max$-function in cases 2(a) and (b) dispensable, because then $X_t-k>0$ always holds. But to preserve the analogy to the usual CUSUM chart, we decided to keep the $\max$-function.

2 If h would be irrational or have a different denominator as k, then $C_t=0,1/n,2/n,\dots ,l$, where $l=m/n<h<(m+1)/n$ with $m,n\in \mathbb{N}$.

3 The same observations concerning Poisson (INAR(1)) CUSUM charts under rising autocorrelation were made in Weiß and Testik (2009).

4 I(h, k) also contains some impossible pairs by construction. As an example, if $$x=c+k$$ with c > 0, then “Scheme (Basic SC_S chart)” section implies the value 0 for the previous CUSUM statistic; but if $$x<k_{\mathrm{W}}$$ at the same time, then CUSUM monitoring will not start at all, making (x, c) an impossible state.