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343 – Advances in Statistical Process Control and Capability

Conditional Median Run Length as Performance Measure of Shewhart S² Control Chart with Estimated Parameter

at Pontifical Catholic University of Rio de Janeiro (PUC-Rio)

Several recent works, in Statistical Monitoring Process, have addressed the effect of parameter estimation on the performance of control charts. Although the measures of this performance are generally expressed based on particular associated characteristics of the unconditional run length (????) distribution, recently, some researchers have proposed the use of the conditional ???? distribution that is considered more meaningful in real applications. The most traditional performance indicator is the average ???? (??????). Nevertheless, this expected value is not a good representative measure for the center because the ???? distribution is usually right-skewed. Therefore, we study the ??-quantiles of the conditional in-control run length (????????????) of the ??2 and ?? control charts. It is worth to note that these quantiles, the distribution of which is obtained analytically, are random variables when the parameter of the ???? process dispersion is estimated. One specific quantile is proposed: the conditional in-control median run length ???????????? (??????????0.50). We analyze and compare the entire distribution of the ???????????? and the conditional ???? ?????? (????????????). The results show that the variability of the ???????????? is smaller than the ????????????. Moreover, the ???????????? and ???????????? requires similar minimum values of number of Phase I samples (??), to guarantee a specified minimum limit (a percentage of its nominal value) with a high probability. For that reason, the proposed robust ???????????? may be a good alternative performance measure.