Evaluation of run-length distribution for CUSUM charts under gamma distributions

Abstract

Numerical evaluation of run-length distributions of CUSUM charts under normal distributions has received considerable attention. However, accurate approximation of run-length distributions under non-normal or skewed distributions is challenging and has generally been overlooked. This article provides a fast and accurate algorithm based on the piecewise collocation method for computing the run-length distribution of CUSUM charts under skewed distributions such as gamma distributions. It is shown that the piecewise collocation method can provide a more robust approximation of the run-length distribution than other existing methods such as the Gaussian quadrature-based approach, especially when the process distribution is heavily skewed. Some computational aspects including an alternative formulation based on matrix decomposition and geometric approximation of run-length distribution are discussed. Design guidelines of such a CUSUM chart are also provided.

Keywords:

• Average run length
• integral equation
• skewed distributions
• statistical process control
• variance