00248 u746734c78cd2a48e08de5c135957e6

250

Yander Wiel

those of comparable CUSUM schemes, though. See Champ and Woodall (1987) for details.

In the case of a random walk (X. = 1) a step shift in the process results in a single residual with a non-zero mean. Obviously the inertia associated with CUSUMs and EWMAs will only weaken the evidence of a shift. The best monitoring scheme for this case is a Shewhart individuals chart.

If Xe(0,l), the best approach to monitoring for step shiits is not so obvious. Since the Shewhart individuals chart is a special case of both EWMAs and CUSUMs, it seems likely that a continuum of EWMA charts or CUSUM charts could be constructed to give good ARL performance on the continuum of IMAs as X varies from 0 to 1. But it also seems possible that a different kind of scheme could be developed that achieves better ARL performance than both the EWMA and the CUSUM by looking specifically for a pattem of geometrie decay in the residuals. The comparisons in Subsection 3.2 show that the first of these hunches is true for the CUSUM chart. To be specific, for a given X. and shift size p., a CUSUM chart can be designed to give ARL performance that is at least as good as the other schemes.

Four monitoring schemes ARL performance of four different classes of monitoring schemes is studied below. The classes are CUSUMs, EWMAs, Shewhart individuals schemes and schemes that use likelihood ratio statistics. Each of these classes is described below and some guidance is given for choosing a particular scheme from within each class based on ARL properties. We also give aids for determining the control limit that results in a given ARL when no shift occurs. This determination is very important when several schemes are being compared because control limits for the schemes should be selected so they have equal ARLs when no shift occurs. We use ARL₀ to

denote this so-called “in control” ARL.

In the descriptions below a, refers to forecast errors calculated using

(1). Nominally, they should approximate an iid N(0, a^{1 2}) sequence. Since we do not consider the effects of estimating a, we assume that a = 1. This is equivalent to assuming that the residuals have been scaled by divided through by ct.

1

Shewhart individuals scheme: This simply signals when |a,| exceeds an

2

action limit h. The action limit can be set to give a desired ARL when no shift occurs. To do this take h to be the 1-1/(2ARL) ąuantile of the standard normal distribution.