Run-length distributions ofQ-chart schemes

Abstract

This article studies the performance of the Shewhart chart of Q statistics in the detection of process mean shifts in start-up processes and short runs. We propose an accurate, analytic approximation of this chart's run-length distribution. Our study reveals that the chart has an early detection advantage in that it is more likely than other methods to detect a process mean shift within the first few observations following the shift. This is a desirable property because early detection should make it easier to identify the cause of the shift, increasing the rate of continuous quality improvement. In addition, our analysis illustrates the importance of reacting immediately to out-of-control signals from the chart as compared to waiting for subsequent observations to confirm the presence of a shift.

Performance and properties of Q‐Statistic monitoring schemes

“…In this article, v is assumed to be fixed, but several different values of v are considered in our numerical studies. PROOF: Follows immediately from Brown et al [2] and Appendix B of Zantek [25].…”

“…Zantek [25] also gives approximate expressions for the probabilities of RLs greater than one, but he finds cases where they appear to understate the true probabilities by a few percentage points (and in percentage terms by more than 10%). In addition, the accuracy of his expressions is not known for all cases.…”

“…Another exact expression for the RL probability in (15) is given by Zantek [25]. Zantek [25] also gives approximate expressions for the probabilities of RLs greater than one, but he finds cases where they appear to understate the true probabilities by a few percentage points (and in percentage terms by more than 10%).…”

“…The performance of a statistical monitoring scheme is typically measured by the run length (the number of observations until the monitoring scheme detects the process change (e.g., mean shift)); see, for example, Hawkins [6], Shu et al [20], Vardeman and Ray [22], Wardell et al [24], or Zantek [25]. The distribution of the run length (RL) is used in designing the scheme and in comparing its performance with that of competing schemes.…”

Shewhart and EWMA t control charts for short production runs

“…Quesenberry 7,8 has defined a set of sequential Q statistics to be used in Shewhart, EWMA and CUSUM charts for the detection of changes in the process mean or variance. Del Castillo and Montgomery 9 and Zantek 10 have investigated the statistical properties of Q charts and proposed some enhancements. He et al 11 have demonstrated that the Q charts with unknown variances are biased: that is, for some specific shift mean sizes the Q charts have a larger out-of-control average run length than the expected number of false alarms.…”