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 . Then for,
e.g.,
and
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
the worse the accuracy of
the Markov chain approach. Therefore,
is set greater for
(
) than for
(
). 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
(see Subsection 11.3.2).
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The corresponding
XploRe
results by using the quantlet
XFGlucsac.xpl
coincide with the values of Lucas and Saccucci (1990).
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 is equal to 0.5.
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First, we compare the critical values. By using
XploRe
(
XFGcrosc.xpl
)
with
one gets
(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
XFGcrosarl.xpl
(
) it turns out, that
again the ARL values coincide with one exception only, namely
for the old scheme with
.
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.
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A slight modification of the above quantlet
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
is set to 25 for the scheme called old scheme by Crosier. The results are
summarized in Table 11.4.
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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 . One further in-control ARL difference one notices
for the new scheme with
. All other differences are small.
There are different sources for the deviations: