Page 396 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

OTE/SPH
OTE/SPH
August 31, 2006
Char Count= 0
3:9
JWBK119-25

25

CUSUM and Backward CUSUM

for Autocorrelated Observations

L. C. Tang and O. O. Atienza

In this chapter we present a backward CUSUM (BCUSUM) scheme based on ideas
from the forecasting literature. A uniformly most powerful test for detecting level
shifts in the mean is derived for this scheme. An equivalent CUSUM scheme based
on its mirror image is shown to have to a parabolic control boundary. The proposed
scheme is further expounded for monitoring autocorrelated data. The parameter for
determining the control limits can be selected based on the desired ARL. Examples
are given to illustrate the idea and the application of these schemes.

25.1 INTRODUCTION

One disadvantage in using the V-mask CUSUM scheme or its tabular counterpart
1
(see chapter 24) is that it is insensitive to large process changes. Lucas noted that
this is related to the anomaly of Wald’s sequential likelihood ratio test on which the
CUSUM scheme is based. He observed that the V-mask constructed based on Wald’s
test is not uniform against other alternatives. This prompted him to modify the V-
2
mask and suggest a semiparabolic mask. Independently, Bissell proposed a similar
3
semiparabolic mask. Rowlands et al., on the other hand, proposed a snub-nosed
V-mask which comprises superimpositions of several V-masks. These modifications
were done to make the CUSUM more uniform in detecting shifts in the mean of normal
random variables.
In designing a control rule for CUSUM, the purely parabolic mask has always been
4
mentioned as a candidate. For example, Barnard noted that some of his colleagues
were using the parabolic mask but gave no further details. Van Dobben de Bruyn 5
also noted that there were reasons to believe that segments of parabolas would be

Six Sigma: Advanced Tools for Black Belts and Master Black Belts L. C. Tang, T. N. Goh, H. S. Yam and T. Yoap
C 2006 John Wiley & Sons, Ltd
381