Moving range charts for monitoring the Weibull shape parameter with single observation samples

Abstract

This paper proposes an approach to monitor shifts in the Weibull shape parameter b via control charts based on the moving range of single-point samples from a smallest extreme value distribution. The average run length (ARL) of the proposed charts are computed using Fredholm integral equations of the second kind. The derived control limits for one-sided and two-sided control charts are unbiased in the sense that the ARL when b has shifted is shorter than the desired stable-process ARL. These control limits depend only on the desired stable-process ARL and the stable value of b. The paper also discusses the sample size requirements for Phase I so that the run length distributions are similar under standards-given scenario (b is given) and retrospective scenario (b is estimated from past data). The proposed methods are then applied to data on the breaking strengths of carbon fibers. The results suggest that one-sided control charts can detect small shifts in b sooner than two-sided charts.

Moving range EWMA control charts for monitoring the Weibull shape parameter

“…The variance given by Equation (2) in [8] is incorrect, and the correct value is given above. We have the following observations:…”

“…The proposed control scheme of Pascual and Nguyen [8] performs well in monitoring the Weibull shape parameter β, when shifts in β are large. However, for small-to-moderate changes, the chart does not signal out-of-control as early as possible.…”

“…, m andR/[2 log(2)], whereR = m i=1 R i /m. • As noted in [8], R i has a scale distribution with scale parameter σ , and R i /σ = βR i is a pivotal quantity.…”

“…The moving range chart proposed by Pascual and Nguyen [8] suits cases when a periodic sample is restricted to a single observation. The proposed control scheme of Pascual and Nguyen [8] performs well in monitoring the Weibull shape parameter β, when shifts in β are large.…”

“…The control limits for one-sided (lower-sided and upper-sided) and two-sided charts are then determined by the desired stable-process ARL and the stable-process value of β, but not the Weibull scale parameter α. Furthermore, we examine ARL properties of the proposed charts and compare the performance of moving range EWMA charts to that of the moving range chart of [8]. We also discuss the sample size requirements for Phase I of retrospective charts.…”