Performance of EWMA versus last observation for feedback control

Two forms of feedback regulation that have been used for process adjustment in Automatic Process Control are considered. In the first, recommended originally by Box and Jenkins (1963) and by Box, Jenkins and MacGregor (1974), action is taken when the absolute difference between an exponentially weighted moving average (EWMA) of the past data [Zcirc]n+1 and the target value T first crosses a threshold value L* In the second, recommended by Taguchi (1981), the action is triggered when the absolute difference between the last observation zn and the target value T exceeds a given constant L. It has been shown (Box and Jenkins 1963) that if we consider the disturbance as an integrated moving average IMA(0,1,1) model containing parameter θ, then the former is optimal when θ is equal to the smoothing parameter used in the EWMA, and the latter is optimal if θ is zero. Thus some controversy has arisen because, as we show here, for any given scheme of the second kind with parameter L, there is a scheme of the first...