Some simplified Shewhart-type distribution-free joint monitoring schemes and its application in monitoring drinking water turbidity

Abstract

In this article, we propose three nonparametric schemes based on the combination of p-values for simultaneous monitoring of shifts in location and scale parameters of a process. Li and Qiu first introduced the idea of dynamic nonparametric monitoring based on the p-value in the context of location shift of a process. This article extends their idea to a more general problem of joint monitoring of location and scale. In this article, we use three approaches for combining the p-values, namely Fisher, Liptak, and Tippett methods, to design three new schemes. We refer to them as the SL-Fisher (hereafter, SLF), the SL-Liptak (hereafter, SLL), and the SL-Tippett (hereafter, SLT) schemes. These schemes are based on a single plotting statistic attained by combining the p-values of the Wilcoxon-rank-sum (WRS) and Ansari-Bradley (AB) test statistics along with the combining functions to control the multiplicity effect. We use a real application based on water turbidity data to explain the implementation of the proposed schemes. We examine and compare the performance characteristics including the mean, median, and several percentiles of the proposed schemes thoroughly using Monte-Carlo simulation. We observe that, in general, the SLT scheme is effective for the detection of shifts in scale followed by either no shift or small shift in the location and the SLL scheme for other cases of shifts. Unlike existing schemes, the proposed schemes offer easier post-signal follow-up approach. We illustrate the design for one-sided shifts in both location and scale, but it can be easily extended for two sided shifts in both location and scale or two-sided shifts in location along with one-sided shifts in scale.