11.3 Comparison with existing methods


11.3.1 Two-sided EWMA and Lucas/Saccucci

Here, we compare the ARL and AD computations of Lucas and Saccucci (1990) with XploRe results. In their paper they use as in-control ARL $ \xi=500$. Then for, e.g., $ \lambda=0.5$ and $ \lambda=0.1$ the critical values are 3.071 and 2.814, respectively. By using XploRe the related values are 3.0712 and 2.8144, respectively. It is known, that the smaller $ \lambda$ the worse the accuracy of the Markov chain approach. Therefore, $ r$ is set greater for $ \lambda=0.1$ ($ r=200$) than for $ \lambda=0.5$ ($ r=50$). Table 11.1 shows some results of Lucas and Saccucci (1990) on ARLs and ADs. Their results are based on the Markov chain approach as well. However, they used some smaller matrix dimension and fitted a regression model on $ r$ (see Subsection 11.3.2).


Table: ARL and AD values from Table 3 of Lucas and Saccucci (1990)
$ \mu$ 0 0.25 0.5 0.75 1 1.5 2 3 4 5
$ \lambda=0.5$
$ {\cal L}_\mu$ 500 255 88.8 35.9 17.5 6.53 3.63 1.93 1.34 1.07
$ {\cal D}_\mu$ 499 254 88.4 35.7 17.3 6.44 3.58 1.91 1.36 1.10
$ \lambda=0.1$
$ {\cal L}_\mu$ 500 106 31.3 15.9 10.3 6.09 4.36 2.87 2.19 1.94
$ {\cal D}_\mu$ 492 104 30.6 15.5 10.1 5.99 4.31 2.85 2.20 1.83


The corresponding XploRe results by using the quantlet 22760 XFGlucsac.xpl coincide with the values of Lucas and Saccucci (1990).

22764 XFGlucsac.xpl


11.3.2 Two-sided CUSUM and Crosier

Crosier (1986) derived a new two-sided CUSUM scheme and compared it with the established combination of two one-sided schemes. Recall Table 3 of Crosier (1986), where the ARLs of the new and the old scheme were presented. The reference value $ k$ is equal to 0.5.


Table: ARLs from Table 3 of Crosier (1986)
$ \mu$ 0 0.25 0.5 0.75 1 1.5 2 2.5 3 4 5
old scheme, $ h=4$  
$ {\cal L}_\mu$ 168 74.2 26.6 13.3 8.38 4.74 3.34 2.62 2.19 1.71 1.31
new scheme, $ h=3.73$  
$ {\cal L}_\mu$ 168 70.7 25.1 12.5 7.92 4.49 3.17 2.49 2.09 1.60 1.22
old scheme, $ h=5$  
$ {\cal L}_\mu$ 465 139 38.0 17.0 10.4 5.75 4.01 3.11 2.57 2.01 1.69
new scheme, $ h=4.713$  
$ {\cal L}_\mu$ 465 132 35.9 16.2 9.87 5.47 3.82 2.97 2.46 1.94 1.59


First, we compare the critical values. By using XploRe ( 22967 XFGcrosc.xpl ) with $ r=100$ one gets $ c=4.0021$ (4), 3.7304 (3.73), 4.9997 (5), 4.7133 (4.713), respectively - the original values of Crosier are written in parentheses. By comparing the results of Table 11.2 with the results obtainable by the quantlet 22970 XFGcrosarl.xpl ($ r=100$) it turns out, that again the ARL values coincide with one exception only, namely $ {\cal
L}_{1.5}=4.75$ for the old scheme with $ h=4$.

22974 XFGcrosarl.xpl

Further, we want to compare the results for the Average Delay (AD), which is called Steady-State ARL in Crosier (1986). In Table 5 of Crosier we find the related results.


Table: ADs (steady-state ARLs) from Table 5 of Crosier (1986)
$ \mu$ 0 0.25 0.5 0.75 1 1.5 2 2.5 3 4 5
old scheme, $ h=4$  
$ {\cal L}_\mu$ 163 71.6 25.2 12.3 7.68 4.31 3.03 2.38 2.00 1.55 1.22
new scheme, $ h=3.73$  
$ {\cal L}_\mu$ 164 69.0 24.3 12.1 7.69 4.39 3.12 2.46 2.07 1.60 1.29
old scheme, $ h=5$  
$ {\cal L}_\mu$ 459 136 36.4 16.0 9.62 5.28 3.68 2.86 2.38 1.86 1.53
new scheme, $ h=4.713$  
$ {\cal L}_\mu$ 460 130 35.1 15.8 9.62 5.36 3.77 2.95 2.45 1.91 1.57


A slight modification of the above quantlet 22979 XFGcrosarl.xpl allows to compute the ADs. Remember, that the computation of the AD for the two-sided CUSUM scheme is based on a twodimensional Markov chain. Therefore the parameter $ r$ is set to 25 for the scheme called old scheme by Crosier. The results are summarized in Table 11.4.


Table: ADs (steady-state ARLs) computed by XploRe , different values to Table 11.3 are printed as italics
$ \mu$ 0 0.25 0.5 0.75 1 1.5 2 2.5 3 4 5
old scheme, $ h=4$  
$ {\cal L}_\mu$ 163 71.6 25.2 12.4 7.72 4.33 3.05 2.39 2.01 1.55 1.22
new scheme, $ h=3.73$  
$ {\cal L}_\mu$ 165 69.1 24.4 12.2 7.70 4.40 3.12 2.47 2.07 1.60 1.29
old scheme, $ h=5$  
$ {\cal L}_\mu$ 455 136 36.4 16.0 9.65 5.30 3.69 2.87 2.38 1.86 1.54
new scheme, $ h=4.713$  
$ {\cal L}_\mu$ 460 130 35.1 15.8 9.63 5.37 3.77 2.95 2.45 1.91 1.57


While the ARL values in the paper and computed by XploRe coincide, those for the AD differ slightly. The most prominent deviation (459 vs. 455) one observes for the old scheme with $ h=5$. One further in-control ARL difference one notices for the new scheme with $ h=3.73$. All other differences are small.

There are different sources for the deviations:

  1. Crosier computed $ D^{(32)}=
({\underline{p}^{32}}^T\underline{L})/({\underline{p}^{32}}^T\underline{1})$ and not the actual limit $ D$ (see 11.8, 11.10, and 11.12).

  2. Crosier used $ ARL(r)=ARL_\infty+B/r^2+C/r^4$ and fitted this model for $ r=8,9,10,12,15$. Then, $ ARL_\infty$ is used as final approximation. In order to get the above $ D^{(32)}$ one needs the whole vector $ \underline{L}$, such that this approach might be more sensitive to approximation errors than in the single ARL case.