Accurate ARL Calculation for EWMA Control Charts Monitoring Simultaneously Normal Mean and Variance

EWMA control charts designed for monitoring the variance or the mean and the variance of a normally distributed variable are either based on the log transformation of the sample variance S or provide only rough ARL results. Gan (1995), as the most prominent example for the simultaneous case, calculated ARL values precisely for X̄-lnS EWMA schemes. The results in Knoth and Schmid (2002) for X̄-S ones are less accurate than the former one. The reason behind the lacking precision is that the methods usually applied for ARL calculation are not able to handle the restricted support of the chart statistic (S and, of course, S and the range R are non-negative random variables). While in Knoth (2005) this problem is treated for single variance monitoring by solving integral equations with collocation methods, this paper employs collocation and ideas similar to Gan (1995) in order to obtain accurate ARL values of X̄-S EWMA control charts. Additionally, the appropriate choice of the non-symmetric control limits for the S part of the scheme is addressed.