Reconciling the Integral Equation and Markov Chain Approaches for Computing EWMA Average Run Lengths

Abstract

The integral equation and Markov chain approaches for computing average run lengths for two-sided exponentially weighted moving average control charts are studied. For the integral equation approach, the choice of numerical method can greatly ease the burden of computation. Gaussian quadrature is recommended when the underlying process data arise from a distribution whose support is the entire real line; however, the Collocation method is to be preferred when the support is finite or semi-infinite. Results for EWMA average run length calculations are given for process data following normal, gamma, t, and uniform distributions. Ultimately, the Markov chain approach is shown to be equivalent to a special case of the integral equation method.

Keywords:

• EWMA charts
• Average run lengths
• Integral equations
• Markov chains

Acknowledgment

The authors would like to thank Dr. Stephen V. Crowder of Sandia National Laboratories for his many insightful comments and suggestions which greatly strengthened this manuscript.