doi:10.1016/j.csda.2005.09.015 
Copyright © 2005 Elsevier B.V. All rights reserved.
Computation of the ARL for CUSUM-S2 schemes
Sven Knoth
, a, 
aAdvanced Mask Technology Center, Postfach 110161, 01330 Dresden, Germany
Received 4 September 2003;
revised 27 September 2005;
accepted 28 September 2005.
Available online 21 October 2005.
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Abstract
Contrary to CUSUM schemes for monitoring the mean of normally distributed random variables, there is a lack of accurate computation of the average run length (ARL) for CUSUM schemes based on the sample variance S2, which are of importance for variance monitoring. Some very accurate methods will be suggested. Evaluating CUSUM charts based on S2 for normal data leads to, naturally, the chi-squared distribution. Then, in the case of even degrees of freedom exact results for Erlang distributed data are employed. For odd degrees piecewise collocation methods are applied for solving the ARL integral equation. Thus, with these methods the ARL for CUSUM-S2schemes can be determined with high precision.
Keywords: Control charts; Average run length; Monitoring variance; Fredholm integral equation of the second kind; Collocation; Erlang distribution
Fig. 1. Accuracy of the Markov chain approach with increasing dimension for high-side CUSUM based on 
, df=1, 
, hh=10, cf. Ramírez and Juan (1989), the true ARL value is 260.7369.
Fig. 2. Comparison of Markov chain approach (BE) and piecewise collocation with increasing dimension for high-side CUSUM based on 
, df=1, 
, hh=10, cf. Ramírez and Juan (1989), the final ARL value is 260.7369.
Fig. 3. Comparison of Markov chain approach (BE), piecewise collocation and Gauss–Legendre Nyström with increasing dimension for high-side CUSUM based on 
, df=5, 
, hh=3.69, cf. Acosta-Mejía et al. (1999), the final ARL value is 200.6695.
Fig. 4. Comparison of Markov chain approach (BE) and piecewise collocation with increasing dimension for low-side CUSUM based on 
, df=1, 
, hl=4.3755, the final ARL value is 100.0004.
Fig. 5. Comparison of Markov chain approach (BE), piecewise collocation and Gauss–Legendre Nyström with increasing dimension for low-side CUSUM based on 
, df=5, 
, hl=2.332, cf. Acosta-Mejía et al. (1999), the final ARL value is 199.6730.
Table 1.
In-control and out-of-control ARL values for high-side CUSUM based on 
, df=1, 
, hh=10, cf. Ramírez and Juan (1989), comparison of piecewise collocation, Markov chain approach (each matrix dimension in parentheses), Hawkins’ anyarl.exe (error code 4, i.e. final value not attained), and Monte-Carlo results (related standard error in parentheses)
Table 2.
In-control and out-of-control ARL values for high-side CUSUM based on S2, df=4, 
, hh=2.921, cf. Chang and Gan (1995), comparison of the original values (by means of a Monte Carlo study, CG1995), the exact results, Markov chain approach, Gauss–Legendre Nyström (each matrix dimension in parentheses), Hawkins’ anyarl.exe, and Monte-Carlo results (related standard error in parentheses)
Table 3.
In-control and out-of-control ARL values for low-side CUSUM based on S2, df=4, 
, hl=1.4457, comparison of the exact results, Markov chain approach, Gauss–Legendre Nyström (each matrix dimension in parentheses), Hawkins’ anyarl.exe, and Monte-Carlo results (related standard error in parentheses)
Table 4.
In-control and out-of-control ARL values for detecting increases in the variance, update of Table 5 of Chang and Gan (1995) with increased hh (from 2.921 to 2.922, confer to Table 2) to ensure an in-control ARL value 100
Table 5.
In-control and out-of-control ARL values for detecting decreases in the variance, update of Table 2 of Acosta-Mejía et al. (1999) (note that their threshold h is linked to our hh by h=df×hh=5×2.332=11.66)
Table 6.
In-control and out-of-control ARL values for detecting increases in the variance, update of Table III of Srivastava (1997), S1997 are the original values of Table III, S1997* are based on (3.6) of Srivastava (1997), SC1992 mark the more precise values à la Srivastava and Chow (1992)
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