One-sided precedence monitoring schemes for unknown shift sizes using generalized 2-of-(h+1) and w-of-w improved runs-rules

Abstract

Parametric monitoring schemes are expected to perform better than their nonparametric counterparts when the assumption of a specific form of a distribution such as the normal is met. In practice, such assumption is often violated. Consequently, the performance of parametric schemes deteriorates considerably. To solve this problem, more efficient and flexible monitoring schemes based on nonparametric tests are recommended. In this paper, efficient and robust one-sided monitoring schemes are developed using generalized {1-of-1 or 2-of-(h+1)} and {1-of-1 or w-of-w} improved runs-rules (IRR) without any distributional assumption in the zero-and steady-states modes. Moreover, the zero-and steady-states run-length properties of the resulting one-sided IRR schemes are formulated and empirically evaluated using the Markov chain technique. Comparisons with other well-known one-sided Shewhart-type nonparametric schemes (e.g. basic precedence scheme and precedence scheme with standard runs-rules) indicate that the proposed schemes have a better performance. Real data are used to demonstrate the design and implementation of the one-sided improved runs-rules precedence schemes. Finally, a discussion on the two-sided IRR precedence schemes is also provided.

Distribution‐free double‐sampling precedence monitoring scheme to detect unknown shifts in the location parameter

“…In Phase II, the process is said to be OOC in stage 1 if the charting statistic or and in stage 2, the process is considered to be OOC if or . The Phase II random variable follows a beta distribution with parameters and in stage 1, and and in stage 2 (see Gibbons and Chakraborti 41 and Malela‐Majika et al 26,27 …”

“…In Phase II, the process is said to be OOC in stage 1 if the charting statistic 𝑌 (𝑗∶𝑛 1 ) ≥ ÔUCL or 𝑌 (𝑗∶𝑛 1 ) ≤ ÔLCL and in stage 2, the process is considered to be OOC if 𝑌 (ℎ∶𝑛) ≥ ÛCL or 𝑌 (ℎ∶𝑛) ≤ LCL. The Phase II random variable 𝑌 follows a beta distribution with parameters 𝑗 and 𝑛 1 − 𝑗 + 1 in stage 1, and 𝑗 and 𝑛 − ℎ + 1 in stage 2 (see Gibbons and Chakraborti 41 and Malela-Majika et al 26,27 ). The conditional probabilities that the charting statistic plots in regions 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 are defined by…”

“…further investigated the 2‐of‐2 SRR and improved runs rules (IRR) precedence schemes based on the minimum and median statistics. More recently, Malela‐Majika et al 26,27 . studied the generalised 2‐of‐( h +1) and w ‐of‐ w one‐ and two‐sided SRR and IRR precedence schemes (with integers 0 and 1).…”

“…Thereafter, Malela-Majika et al 25 further investigated the 2-of-2 SRR and improved runs rules (IRR) precedence schemes based on the minimum and median statistics. More recently, Malela-Majika et al 26,27 studied the generalised 2-of-(h+1) and w-of-w one-and two-sided SRR and IRR precedence schemes (with integers ℎ > 0 and 𝑤 > 1). The focus of this paper is on the Shewhart-type precedence monitoring schemes.…”

Nonparametric precedence chart with repetitive sampling

“…When this assumption is violated, the use of parametric tests will lead to unreliable results and invalid inference; see, for example, Chakraborti et al (2004). To remedy this problem, the literature recommends either the use of data transformation such as the log or square transformation (see Feng et al, 2012) or nonparametric tests such as the Wilcoxon rank‐sum, sign, signed‐rank, precedence and exceedance (see Human et al, 2010; Malela‐Majika et al, 2021, 2022). The former technique does not allow the interpretation of the results based on the original data.…”