The conventional cumulative sum (CUSUM) withk= 0.5 is often used as the default CUSUM statistic when future shifts are unknown. In this paper, CUSUM procedures are designed to be efficient at signalling a range of future expected but unknown location shifts. Two approaches are advocated. The first uses three simultaneous conventional CUSUM statistics with different resetting boundaries. This results in a procedure that has, on average, several levels of memory, and thus signals a broader range of location shifts more efficiently than the conventional CUSUM withk= 0.5. The second uses an adaptive CUSUM statistic that continually adjusts its form to be efficient for signalling a one-step-ahead forecast in deviation from its target value. Average run length (ARL) is used to compare the relative performance of procedures. Several applications are used to illustrate procedures.

Keywords: average run length, exponentially weighted moving average, robst control charts

*by* **Ross S. Sparks, CSIRO, Locked Bag 17, North
Ryde, NSW 2113, Australia**

*INTRODUCTION*

Applied statisticians and engineers implementing monitoring procedures for their processes are often burdened with making the choice between several competing charting methods. Advice given in the literature is: use a Shewhart chart to signal large shifts effciently and either a cumulative sum (CUSUM) or an exponentially weighted moving average (EWMA) chart to signal smaller shifts effciently. Often practitioners wish to signal a range of expected shifts effciently. Lucas (1982) suggested the simultaneous use of the conventional CUSUM and Shewhart charts as a possible solution. This paper explores several alternative methods that are effcient for signalling a range of location shifts.

All methods considered in this
paper are based on Page (1954)´s CUSUM, which has proven
an effective alternative to Shewhart charts for monitoring
industrial processes (Montgomery (1991)). Assume observations
*y _{t}* are independent and identically distributed
(i.i.d.) normal with mean equal to the target value

and

respectively. Usually,
= 0 and
= 0. The appropriate one-sided conventional CUSUM statistic
with *k* = ||/2
is optimal for detecting a location shift for *y _{t}*
of
from target (see Moustakides (1986)). If the location shifts
are always equal to
with
known in advance, then we can optimize the performance of
the one-sided conventional CUSUM. In reality we never know
the magnitude of future shifts, and shifts generally vary
in magnitude. Therefore, a CUSUM statistic is needed that
performs well on average over a range of expected location
shifts.

Where the magnitude of future
shifts is unknown, the conventional CUSUM with *k* =
0:5 is often applied. Often
and
are simultaneously plotted below and above the zero (horizontal)
axis, respectively, in a time sequence plot. For a suitable
*h* > 0, a signal is given whenever
> *h* or
< –*h*.

In this paper, CUSUM charts are developed that are robust at signalling a range of small location shifts. Particular emphasis is given to adapting the conventional CUSUM procedure to achieve this aim. This is achieved using two approaches:

- The first is to apply several CUSUM charts simultaneously
with different levels of memory by using CUSUM statistics
with different
*k*values. A signal is given if any one of these CUSUM statistics exceeds their respective boundary. The next section of this paper introduces the use of several simultaneous CUSUM statistics. - The second, an adaptive CUSUM procedure, is useful in
situations where effcient one-step ahead forecasts of location
changes
can be made. This one-step ahead forecast of the location
change
is used to adapt the CUSUM statistic to be approximately
*locally*optimal for detecting location shifts from the target. The paper uses the simple exponentially weighted moving average to make these one-step ahead forecasts. The third section of the paper outlines the adaptive CUSUM procedure.

The nature of all possible location shifts is infinite, but most are similar to an impulse, a step change or a ramp change. An impulse shift, large step changes and large ramp changes are best detected using a Shewhart chart (Montgomery (1991)). Small step changes and small ramp changes, however, are more quickly signalled using CUSUM charts than Shewhart charts. The first part of the fourth section compares CUSUM procedures for step-changes, while the second part compares them for ramp changes. Practical applications are considered where the changes may vary in nature and magnitude. The paper finishes with some concluding remarks and a step-by-step guide on how to use the adaptive CUSUM.

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