A nonparametric control chart based on the Mann-Whitney statistic

Abstract

Nonparametric or distribution-free charts can be useful in statistical process control problems when there is limited or lack of knowledge about the underlying process distribution. In this paper, a phase II Shewhart-type chart is considered for location, based on reference data from phase I analysis and the well-known Mann-Whitney statistic. Control limits are computed using Lugannani-Rice-saddlepoint, Edgeworth, and other approximations along with Monte Carlo estimation. The derivations take account of estimation and the dependence from the use of a reference sample. An illustrative numerical example is presented. The in-control performance of the proposed chart is shown to be much superior to the classical Shewhart $\bar{X}$ chart. Further comparisons on the basis of some percentiles of the out-of-control conditional run length distribution and the unconditional out-of-control ARL show that the proposed chart is almost as good as the Shewhart $\bar{X}$ chart for the normal distribution, but is more powerful for a heavy-tailed distribution such as the Laplace, or for a skewed distribution such as the Gamma. Interactive software, enabling a complete implementation of the chart, is made available on a website.