Designing X¯ charts for known autocorrelations and unknown marginal distribution

In the design of the X¯ control chart, both the sample size m   of X¯ and the control-limit factor k (the number of standard deviations from the center line) must be determined. We address this problem under the assumption that the quality characteristic follows an autocorrelated process with known covariance structure but unknown marginal distribution shape. We propose two methods for determining m and k, chosen to minimize the out-of-control ARL (average run length) while maintaining the in-control ARL at a specified value. Method 1 calculates the ARL values as if the sample means were independent normal random variables; Method 2 calculates the ARL values as if the sample means were an AR(1) process. Method 2 outperforms Method 1 when the correlation and mean shift are both high. We also modify Methods 1 and 2 with a minimum sample size of 30; the modification moves the in-control ARL closer to the specified value. Our numerical results show that the modified Method 2 performs better than two previous design procedures, especially when the correlation is high.