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CUSUM Scheme for Autocorrelated Observations 395
that has a much larger ARL than expected. Conversely, when φ is underestimated,
the resulting parabola has a narrower envelope than expected. This will result in
a smaller than expected ARL. This property of the proposed CUSUM is similar to
the properties of the CUSUM or exponentially weighted moving average applied to
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forecast residuals ).

25.4.3.2 ARL comparisons
For various levels of autocorrelation φ 1 and shift in mean δ, Atienza et al. 28 compare
the ARL performance of the proposed CUSUM with the SCR and CUSUMR. Their
results are replicated in Table 25.4. The shift in mean is measured in terms of both σ ε
and σ y . For comparison purposes, the control chart parameters for SCR, CUSUMR
and the proposed CUSUM are chosen such that the in-control ARL for the one-sided
scheme is approximately 740.
From Table 25.4, it can be seen that the performance of the proposed CUSUM is
very close to the performance of the CUSUM on residuals. For large process shifts,
the proposed CUSUM outperforms both the SCR and CUSUMR. In detecting small
shifts for highly positively autocorrelated processes, CUSUMR is more sensitive than
the proposed CUSUM. Note, however, that for φ 1 close to 1 (i.e. highly positively
autocorrelated processes) a process shift measured in term of σ ε is small compared
to when the shift is measured in terms of σ y . For example, a 2σ ε shift for an AR(1)
process with φ 1 = 0.75 is equal to only a 1.323 σ y shift. Similarly, when σ 1 = 0.9,a2σ ε
shift in mean is equivalent to only a 0.872 σ y shift. By this token, we can expect that the
proposed CUSUM will outperform CUSUMR in a wider shift range when the shift is
measured in terms of σ y .

25.4.3.3 Length of the mask arm
Ideally, in implementing the CUSUM in a mask scheme, one needs to extend the mask
arms to cover all the data points. Thus, when the process is stable, we will often need to
store a large amount data for process monitoring. To determine the effect of the length
of mask arms on the sensitivity of the proposed CUSUM in detecting a shift in mean,
Atienza et al. 28 analyze the performance of the CUSUM when the analysis is based
only on the latest mw (moving window) of process observations. For a particular mw
size, the value of z * is adjusted such that the in-control ARL will be approximately
740. The results are taken from Atienza et al. 28 and shown in Table 25.5. Note that the
ARL figures in the table are calculated using the same approach as described in the
previous section.
From Table 25.5, we can see that for small values of φ 1 , we may focus our attention
on the latest 50 observations without severely compromising the sensitivity of the
CUSUM scheme. For larger values of φ 1 , we need to concentrate on the latest 100
observations. For all cases, the performance of the CUSUM scheme utilizing only the
latest 200 observations is already quite close to the performance of the CUSUM that
uses all process observations. From these results, it is quite clear that one does not
need to indefinitely extend CUSUM mask arm in order to effectively detect shifts in
process mean.