Self-Starting Cusum Charts for Location and Scale

Abstract

In some quality control problems, it is not known what the exact process mean and standard deviation are under control but it is desired to determine whether there have been drifts from the conditions obtained at the process start-up. This situation is not well-covered by standard cumulative sum procedures, which generally assume known process parameters. This paper uses the running mean and standard deviation of all observations made on the process since start-up as substitutes for the unknown true values of the process mean and standard deviation. Using some theoretical properties of independence of residuals, two pairs of cusums are set up: one testing for constancy of location of the process, and the other for constancy of the spread. While the process is under control, both these cusum pairs are of approximately normal $N(0, 1)$ quantities (and therefore are well understood), but if the location, the spread or both change, then non-centrality is introduced into one or both of the location and scale cusum pairs, and it drifts out-of-control. It is shown that the procedure performs well in detecting changes in the process, even in comparison with the often utopian situation in which the process mean and variance are known exactly prior to the start of the cusum.