Sequential rank cumulative sum charts for location and scale

PhD (Risk Analysis), North-West University, Potchefstroom Campus

In this thesis we construct CUSUMs based on the signed and unsigned sequential ranks of independent observations for the purpose of detecting either a persistent location or a persistent scale shift. In designing these CUSUMs we consider two scenarios, namely detecting a shift when the in-control distribution is symmetric around a known median and when either the symmetry assumption fails or the in-control median is unknown. We then extend our CUSUM designs to the class of Girschick-Rubin CUSUMs. All of our CUSUMs are distribution free and fully self starting: no parametric speci cation of the underlying distribution is necessary in order to finnd correct control limits that guarantee a speci ed nominal in-control average run length given a reference value. In particular, our sequential rank CUSUMs have zero between-practitioner variation. Furthermore, these CUSUMs are robust against the e ect of spurious outliers. The out-of-control average run length properties of the CUSUMs are gauged qualitatively by theorybased calculations and quantitatively by Monte Carlo simulation. We show that in the case where the underlying distribution is normal with an unknown variance, our sequential rank CUSUMs based on a Van der Waerden-type score can be used to good e ect, because the out-of-control average run lengths correspond very well to those of the standard normal distribution CUSUM where the variance is assumed known. For heavier tailed distributions we show that use of the Wilcoxon sequential rank score is indicated. Where transient special causes are apt to occur frequently, use of a Cauchy score is indicated. We illustrate the implementation of our CUSUMs by applying them to data from industrial environments.

Doctoral