The existing procedures for designing exponentially weighted moving average (EWMA) control charts are based on the assumption of known process parameters. In practice, these parameters are usually unknown and replaced with estimates from an in-control reference sample. Using parameter estimates with design procedures intended for known parameters can lead to significantly deteriorated chart performance. In this paper, the assumption of known parameters is relaxed and design procedures for the EWMA chart are developed accordingly.Key Words: Average Run Length, Exponentially Weighted Moving Average, Integral Equation.

*By* **L. ALLISON JONES, University of Miami, Coral
Gables, FL 33124-8237
**

Introduction

IN most industrial and service applications, the mean and
standard deviation of the process to be monitored are unknown.
A common practice is to estimate the parameters of the process
from an in-control (IC) reference sample and to establish
the control chart using these estimates. Most statistical
procedures for designing control charts are based on the sampling
distribution of the plotted statistics, and are developed
assuming known parameters. When estimates are used in place
of known parameters, the appropriate sampling distribution
of the chart statistics should account for the variability
in the estimators; otherwise, the IC and out-of-control (OC)
performance of the charts can be strongly affected. Failing
to account for parameter estimates when designing exponentially
weighted moving average (EWMA) charts can lead to an increase
in the number of false alarms and a reduction in the ability
of the charts to detect process changes. Small reference sample
sizes magnify the adverse effects of estimation. For example,
when parameter estimates from *m* = 30 IC subgroups of
size *n* = 5 are used in an EWMA monitoring scheme with
a smoothing constant of 0.1, the probability of a false alarm
within twenty observations is in ated by nearly 110%. This
is a rarely mentioned and unacceptable side effect of substituting
estimates into the existing EWMA design procedures.

One solution to this problem is to
simply increase the size of the reference sample in order
to reduce the variability in the sampling distributions of
the estimates. Although a widely accepted heuristic is that
*m* = 30 subgroups from a process usually produce reasonable
estimates, Quesenberry (1993) suggested that at least *m*
= 100 subgroups of size *n* = 5 are required to adequately
estimate the parameters when designing an
chart. Jones, Champ, and Rigdon (2001) showed that the sample
size necessary to achieve expected statistical performance
when designing an EWMA chart can be much larger than *m*
= 100. The sufficient sample size depends on the EWMA smoothing
constant, with larger samples required when smaller smoothing
constants are used. When smoothing constants of 0.1 or less
are used, as many as 400 IC subgroups are necessary to estimate
parameters (Jones et al. 2001). In some applications where
data are plentiful, it might be feasible to wait until 400
IC subgroups have accumulated in order to begin monitoring
the process. In many situations, however, waiting may prove
detrimental to the quality of the product or service that
is being monitored.

If large sample sizes are not available, the practitioner is left with a dilemma. The EWMA chart can be used with estimates based on small samples, but the statistical performance of the chart may be poor; that is, the chart may signal frequently with no assignable causes present. However, if process monitoring is delayed to obtain the necessary data, expensive process problems may go undetected during this time. Another approach is to use a self-starting procedure, such as the Q-chart, which is designed with the assumption that parameters are estimated (Quesenberry 1995). Self-starting charts can be used for monitoring earlier than traditional control charts because the parameter estimates are updated with the addition of each new observation. One limitation of self-starting procedures is the "masking", or parameter adaptation, problem. If an early process change is not quickly detected, then the parameter estimates may be adversely a ected by the change, thus masking the shift from future detection.

The objective of this paper is to develop design procedures for the EWMA chart that do not require the assumption of known parameters. The new procedures give wider control limits by reflecting the variability of the parameter estimates used in the sampling distribution of the EWMA chart statistics. Charts developed with the new control limits show improved IC performance since they are designed to achieve a speci ed IC average run length (ARL).

The literature regarding the EWMA chart is substantial. The next section gives a very brief review of articles pertaining to EWMA control chart design. Following this, some background information regarding the EWMA model is presented. The new design procedures for the EWMA chart with estimated parameters are given next. An example illustrating the application of the new design procedures follows. A performance comparison between the existing design methods and the new method is given. Finally, some concluding remarks are made.

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