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

Abstract

Exponentially weighted moving average (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 2 or provide only rough average run length (ARL) results. Gan (1995), as the most prominent example for the simultaneous case, calculated ARL values precisely for -ln S 2 EWMA schemes. The results in Knoth and Schmid (2002) for -S 2 ones are less accurate than the former one. The reason behind the lack of precision is that the methods usually applied for ARL calculation are not able to handle the restricted support of the chart statistic (S 2 and, of course, S and the range R are nonnegative 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 -S 2 EWMA control charts. Additionally, the appropriate choice of the nonsymmetric control limits for the S 2 part of the scheme is addressed.