Accurate ARL computation for EWMA-S2 control charts

Abstract

Originally, the exponentially weighted moving average (EWMA) control chart was developed for detecting changes in the process mean. The average run length (ARL) became the most popular performance measure for schemes with this objective. When monitoring the mean of independent and normally distributed observations the ARL can be determined with high precision. Nowadays, EWMA control charts are also used for monitoring the variance. Charts based on the sample variance S 2 are an appropriate choice. The usage of ARL evaluation techniques known from mean monitoring charts, however, is difficult. The most accurate method-solving a Fredholm integral equation with the Nyström method-fails due to an improper kernel in the case of chi-squared distributions. Here, we exploit the collocation method and the product Nyström method. These methods are compared to Markov chain based approaches. We see that collocation leads to higher accuracy than currently established methods.

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

“…The same could be observed in papers about ARL computation for single variance control charts (see Knoth, 2005 for more details). In Knoth and Schmid (2002) the combined X-S 2 EWMA scheme was analyzed, but with less accuracy than Gan (1995) provided.…”

“…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 X-S 2 EWMA control charts. Additionally, the appropriate choice of the nonsymmetric control limits for the S 2 part of the scheme is addressed.…”

Risk-Adjusted Cumulative Sum Charting Procedure Based on Multiresponses

“…Knoth (2005) demonstrated that the collocation method is fast and accurate in calculating the ARL of an exponentially weighted moving average control chart for monitoring the variance of normally distributed data. Consider a CUSUM chart obtained by plotting C n = max(0, C n−1 + W n ), against the sample number n where C 0 = u, 0 ≤ u < h and W n is the log-likelihood ratio statistic…”

“…To solve for c j 's, we choose a set of N nodes in the domain [0, h], and then solve the resulting system of linear equations as discussed in Hackbusch (1995). According to Knoth (2005), the Chebyshev polynomials T j (z) = cos((j − 1) arccos(z)), z ∈ [−1, 1], j = 1, . .…”

The economic design of multivariate binomial EWMA VSSI control charts

“…[30] compared the performances of the Markov chain approach and the simulation method to approximate ARL and showed that there was no significant difference between them. Knoth [14] did a comparison between Markov chain and integral equation approaches in terms of ARL evaluation of EWMA-S 2 control charts. Rigdon [29] applied the integral equation method for the calculation of in-control ARL (ARL 0 ) and out-of-control ARL (ARL 1 ) for the MEWMA charts.…”