On Statistical Design of theS²Control Chart

Abstract

The usual S 2 chart is constructed with equal false alarm probabilities for the lower and upper control limits. An alternative is the ARL-unbiased S 2 chart, where the ARL (average run length) curve attains its maximum when the common-cause variance is at its in-control value. We examine the ARL properties of these S 2 charts, and show that neither the usual S 2 chart nor the ARL-unbiased S 2 chart is optimal in a natural sense. We propose two criteria, under which optimal S 2 charts can be constructed. The effects of parameter estimation on our designs are also discussed, and suggestions are made regarding practical implementation.

On ARL-Unbiased Control Charts

“…The next proposition provides an expression for the argmax of ARL .Â/; this result and its proof were adapted from Morais (2002, Example 2.6, pp. 20-22) and can also be found in Zhang et al (2005) and Uhlmann (1982, p. 213). …”

“…This fact is responsible for a serious and well-known limitation: Shewhart charts are not effective in the detection of small to moderate changes in the parameter being monitored. This limitation led to the proposal of alternative charts, such as the 3 Regretfully, this chart has not uniformly smaller out-of-control ARLs than any other S 2 -chart, as noted by Zhang et al (2005). This result prompted these authors to propose two alternatives to the ARL-unbiased S 2 -chart.…”

Optimal design of a synthetic chart for monitoring process dispersion with unknown in-control variance

“…Notice that Tðr 0 Þ ¼ qTðr 1 Þ and Tðr 0 Þ is an increasing function of q, thus Tðr 0 Þ has been proposed as a monitoring statistic (Montgomery, 2005;Guo & Wang, 2014;Zhang et al, 2005).…”

“…When L ! 1, the synthetic S 2 chart reduces to an ARL-unbiased S 2 chart proposed by Zhang et al (2005).…”

“…In Section 2, we introduce the synthetic S 2 chart with known parameter and compare the performance of the proposed synthetic S 2 chart with the ARL-unbiased S 2 chart proposed by Zhang, Bebbington, Lai, and Govindaraju (2005) in terms of the run length properties. In Section 3, we study the effect of parameter estimation on the proposed synthetic chart.…”