On the design of nonparametric runs‐rules schemes using the Markov chain approach

Abstract

The aim of this paper is to highlight some concerns about the partly inaccurate manner in which the currently available 2‐of‐2 and 2‐of‐3 simple and improved runs‐rules charts based on the sign and the signed‐rank statistics were designed using Markov chain matrix. Because of the memory‐less property of the Markov chains, the empirical average run‐length (ARL) values were not affected; but the design structure of the matrix makes it difficult to formulate the general expressions of the ARL explicitly. Also, the dimension (and consequently, the simplicity) of the transition probability matrices (TPMs) and the false alarm rates expressions were affected negatively. Thus, we present some zero‐ and steady‐state formulae that make it easier to construct some key elements of the run‐length distribution (including the TPMs) of the simple and improved 2‐of‐(H + 1) runs‐rules charts based on the sign and the signed‐rank statistics, for any positive integer value H > 0, not just H = 1 and 2, as currently available for these monitoring schemes, so that any interested reader may use these formulae to obtain the corresponding empirical study.

Robust Distribution-Free Hybrid Exponentially Weighted Moving Average Schemes Based on Simple Random Sampling and Ranked Set Sampling Techniques

“…ese rules are known as runs-rules (or stopping rules). For more details on the different types of runs-rules, readers are referred to Klein [20], Khoo and Ariffin [21], Antzoulakos and Rakitzis [22], and Shongwe [23] as well as Adeoti and Malela-Majika [24]. In this paper, the standard runs-rules (SRR) and the improved runs-rules (IRR) approaches are considered.…”

An extended nonparametric homogeneously weighted moving average sign control chart

“…proposed the HWMA chart based on the sign and signed‐rank statistics for monitoring the location parameter. Shongwe 29 studied the design of nonparametric sign and signed‐rank charts with runs rules using the Markov chain approach. Aslam et al 30 .…”

“…Raza et al 28 proposed the HWMA chart based on the sign and signed-rank statistics for monitoring the location parameter. Shongwe 29 studied the design of nonparametric sign and signed-rank charts with runs rules using the Markov chain approach. Aslam et al 30 developed a modified EWMA sign chart while Alevizakos et al 31 proposed the TEWMA sign chart which is more effective than the CUSUM, GWMA, and DEWMA sign charts for small shifts.…”

Treating Measurement Errors in the Run Rule Schemes Integrated with Shewhart X ¯ Chart

“…en, run rules combined with control charts were studied by many researchers, such as Divoky and Taylor [2]; Champ and Woodall [3]; Klein [4]; Fu et al [5]; and so on. More recent works on control charts integrated with run rule schemes can be found in Hu and Castagliola [6]; Chew et al [7]; Shongwe and Graham [8]; Shongwe and Graham [8]; Shongwe et al [9]; Khaw and Chew [10]; Chew et al [11]; Shongwe [12]; Malela-Majika and Rapoo [13]; Malela-Majika et al [14]; Hu et al [15]; and Ali et al [16]. All these existing research studies focused on how to improve the chart's performance by incorporating run rules.…”