Monitoring Weibull Quantiles by EWMA Charts Based on a Pivotal Quantity Conditioned on Ancillary Statistics

Bizuneh

2017

In this paper, we propose 3 new control charts for monitoring the lower Weibull percentiles under complete data and Type‐II censoring. In transforming the Weibull distribution to the smallest extreme value distribution, Pascaul et al (2017) presented an exponentially weighted moving average (EWMA) control chart, hereafter referred to as EWMA‐SEV‐Q, based on a pivotal quantity conditioned on ancillary statistics. We extended their concept to construct a cumulative sum (CUSUM) control chart denoted by CUSUM‐SEV‐Q. We provide more insights of the statistical properties of the monitoring statistic. Additionally, in transforming a Weibull distribution to a standard normal distribution, we propose EWMA and CUSUM control charts, denoted as EWMA‐YP and CUSUM‐YP, respectively, based on a pivotal quantity for monitoring the Weibull percentiles with complete data. With complete data, the EWMA‐YP and CUSUM‐YP control charts perform better than the EWMA‐SEV‐Q and CUSUM‐SEV‐Q control charts in terms of average run length. In Type‐II censoring, the EWMA‐SEV‐Q chart is slightly better than the CUSUM‐SEV‐Q chart in terms of average run length. Two numerical examples are used to illustrate the applications of the proposed control charts.

“…Haghighi et al and Pascual et al proposed a monitoring statistic Q based on a pivotal quantity conditioned on ancillary statistics for monitoring the Weibull percentiles, which is given by

Q = \frac{{\overset{falsê}{y}}_{1, p} - y_{1, p}^{0}}{\overset{falsê}{σ}},

where

{\overset{falsê}{y}}_{1, p} = \overset{μ}{true} + \overset{σ}{true} log ([], - log ((), 1 - p))

and

y_{1, p}^{0}

is the stable process 100 p th percentile.…”

Section: Existing Control Chartsmentioning

confidence: 99%

“…The conditional cdf of Q ∣ a for complete and Type‐II censoring is derived by Pascaul et al

F (()|, q, a) = Pr (()|, Q \leq q, a) = \int_{0}^{\infty} h (()|, x, a) G_{r} true(g ((),,,,, a, p, q, ρ_{p}, x) italicdx, - \infty < q < \infty,

where

g ((),,,,, a, p, q, ρ_{p}, x) = e^{[s_{p} - ρ_{p} - x (s_{p} - q)]} ([], \sum_{j = 1}^{r} e^{(a_{j} x)} + ((), n - r) e^{(a_{r} x)}), h (()|, x, a) = \frac{k ((),,, a, n, r) x^{r - 2} e^{[(x - 1) \sum_{j = 1}^{r} a_{j}]}}{{({}, \frac{1}{r} ([], \sum_{j = 1}^{r} e^{({xa}_{j})} + ((), n - r) e^{({xa}_{r})}))}^{r}}, k ((),,, a, n, r) = {\{{normal\int}_{0}^{\infty} \frac{x^{r - 2} e^{([], ((), x - 1) \sum_{j = 1}^{r} a_{j})}}{{[\frac{1}{r} (\sum_{j = 1}^{r} e^{((), {italicxa}_{j})} + (n - r) e^{((), {italicxa}_{r})})]}^{r}} dx\}}^{- 1},

…”

Section: Existing Control Chartsmentioning

confidence: 99%

“…The EWMA statistic E i in the EWMA‐SEV‐Q control chart is established as

E_{i} = ((), 1 - λ) E_{i - 1} + λ Q_{i}, E_{0} = μ ((), a),

where 0 < λ ≤ 1 and μ ( a ) is the mean of Q conditioned on the ancillary a . The fixed control limits are given by

({, \begin{array}{c} h_{L} = μ ((), a) - italicLσ ((), a) \sqrt{\frac{λ}{2 - λ}} \\ h_{U} = μ ((), a) + italicLσ ((), a) \sqrt{\frac{λ}{2 - λ}} \end{array}),

where σ ( a ) is the standard deviation of Q conditioned on the ancillary a .…”

Section: Existing Control Chartsmentioning

confidence: 99%

({, \begin{array}{c} C_{i}^{+} = max ((),, 0, C_{i - 1}^{+} + Q_{i} - k^{+}), C_{0}^{+} = 0 \\ C_{i}^{-} = min ((),, 0, C_{i - 1}^{-} + Q_{i} + k^{-}), C_{0}^{-} = 0 \end{array}),

where k + and k − represent the reference values of the upward and the downward shifts in percentile, respectively.…”

Section: Three New Control Chartsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

New control charts for monitoring the Weibull percentiles under complete data and Type‐II censoring

Bizuneh

2017

An exponentially weighted moving average chart based on likelihood‐ratio test for monitoring Weibull mean and variance with subgroups

2017

Monitoring changes in the Weibull mean and variance simultaneously is of interest in quality control. The mean and variance of a Weibull process are determined by its shape and scale parameters. Most studies are focused on monitoring the Weibull scale parameter with fixed shape parameter or the Weibull shape parameter with fixed scale parameter. In this paper, we propose an exponentially weighted moving average chart based on the likelihood‐ratio test and an inverse error function called ELR chart to monitor changes in the Weibull mean and variance simultaneously. The simulation approach is used to derive the average run length. We compare our proposed chart with other existing control charts for 3 cases, including scale parameter changes with fixed shape parameter, shape parameter changes with fixed scale parameter, and both parameters changes. The results show that the ELR chart outperforms the other control charts in terms of average run length in most cases. Two numerical examples are used to illustrate the applications of the proposed control chart.

EWMA chart based on Bayes‐conditional pivotal quantity for Weibull percentiles under complete data and type‐II censoring

Bizuneh

2018

Bayes-conditional control chart has been used for monitoring the Weibull percentiles with complete data and type-II censoring. Firstly, the Weibull data are transformed to the smallest extreme value (SEV) distribution. Secondly, the posterior median of quantiles is used as a monitoring statistic. Finally, a pivotal quantity based on the monitoring statistic with its conditional distribution function is derived for obtaining the control limits. This control chart is denoted as Shewhart-SEV-e Q Bp . In this study, we extend this work based on an exponential weighted moving average model named exponential weighted moving average-SEV-e Q Bp for monitoring the Weibull percentiles.We provide the statistical properties of the monitoring statistic. The average run length and the standard deviation of run lengths, computed by the integral equation approach, are used as performance measures. The results indicate that the proposed chart performs better than the Shewhart-SEV-e Q Bp . The breaking strength of carbon fibers is used to illustrate the application of the proposed control chart.