A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules

Abstract

a b s t r a c tNonparametric control charts do not require knowledge about the shape of the underlying distribution and can thus be attractive in certain situations. Two new Shewharttype nonparametric control charts are proposed for monitoring the unknown location parameter of a continuous population in Phase II (prospective) applications. The charts are based on control limits given by two specified order statistics from a reference sample, obtained from a Phase I (retrospective) analysis, and using some runs-type signaling rules. The plotting statistic can be any order statistic in a Phase II sample; the median is used here for simplicity and robustness. Exact run length distributions of the proposed charts are derived using conditioning and some results from the theory of runs. Tables are provided for practical implementation of the charts for a given in-control average run length (ARL 0 ) between 300 and 500. Comparisons of the average run length ARL, the standard deviation of run length (SDRL) and some run length percentiles show that the charts have robust in-control performance and are more efficient when the underlying distribution is t (symmetric with heavier tails than the normal) or gamma (1, 1) (right-skewed). Even for the normal distribution, the new charts are quite competitive. An illustrative numerical example is given. An added advantage of these charts is that they can be applied before all the data are collected which might lead to savings in time and resources in certain applications.

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

“…Other nonparametric charts based on runs-type signalling rules have also been proposed in the literature (see e.g. Chakraborti et al (2009)). …”

Distribution-free Phase II Mann–Whitney control charts with runs-rules

“…In Section 5 a performance comparison among the proposed runs-rules enhanced MW charts is done. We also include, in the comparison study, the distribution-free precedence charts proposed by Chakraborti et al [10]. Note that although these authors considered the RR 2-of-2 scheme, they did not consider the improved runs-rules and, accordingly, for the performance comparison we used the improved precedence chart proposed by Malela-Majika et al [16].…”

“…Chakraborti et al [10] showed that the KL scheme outperforms the DR scheme and, accordingly, only the KL scheme is considered in this paper. The 2-of-2 KL runs-rule will be denoted by RR 2-of-2 from this point forward.…”

“…It is meaningful to compare the runs-rule enhanced MW charts with other available nonparametric and RR 2-of-2 Shewhart precedence charts proposed by Chakraborti et al [21] and Chakraborti et al [10], respectively. In the latter papers the N(0,1), GAM(1,1) and t (4) Tables 7, 8 and 9, respectively.…”

“…Koutras et al [9] reviewed the well-known Shewhart-type charts supplemented with additional rules based on the theory of runs and scans. A class of Phase II nonparametric charts based on precedence statistics with runs-rules was proposed by Chakraborti et al [10]. Zhang and Castagliola [11] investigated the runs-rules based Shewhart ̅ charts when process parameters are unknown.…”