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The sample variance of the quality characteristic shows the variation of a product or process. When the precision is a concern, a tolerance interval (TI) for sample variances is needed. In this paper, we construct β-expectation TIs for the population of sample variances, assuming normality of the data, from the frequentist and Bayesian perspectives. The accuracy level of each TI, which is the probability that the actual coverage falls into a specified range, is calculated to evaluate the performance of the TI. Besides, the determination of the minimum number of subgroups and the minimum subgroup size is discussed to achieve the desired accuracy level. When the accuracy level, the number of subgroups and subgroup size are known, the maximum variation is also calculated to measure the goodness of the proposed TI. Comparison results show that a Bayesian TI outperforms the corresponding frequentist TI. A real example is used to demonstrate the applicability and implementations of the proposed TIs. [ABSTRACT FROM AUTHOR]

Studies on control charts with estimated parameters have received much attention in the recent literature. In this paper, the effect of parameter estimation on the performance of the Shewhart-type chart for the Rayleigh distribution, namely the V R chart, is first studied under the conditional perspective. It is found that parameter estimation has a serious effect on the performance of the frequentist V R chart. In order to solve this problem, the frequentist V R chart is adjusted by using the exceedance probability criterion to guarantee the in-control performance. Since the frequentist V R chart uses the sample information from Phase I, but not the process information from past experience, an alternative chart, namely the Bayesian V R chart, is proposed based on the predictive distribution of the plotting statistic. The performances of the Bayesian and adjusted frequentist V R charts are evaluated and compared in terms of the percentiles, mean, and standard deviation of the conditional average run length distribution. The results suggest that the Bayesian chart outperforms the frequentist counterpart, especially when more prior information is available. [ABSTRACT FROM AUTHOR]

Process capability index (PCI) is widely used to evaluate the process capability in various industries. In this article, generalized confidence intervals (GCIs) of two widely used percentile-based PCIs for the inverse Gaussian (IG) distribution are constructed. Meanwhile, the GCI of the difference between two IG processes' PCIs is also considered. In addition, the performance of the proposed GCI is evaluated and compared with that of the confidence intervals based on the traditional bootstrap method. The simulation results indicate that the GCI outperforms the bootstrap counterparts in terms of the coverage probabilities, and the actual coverage probabilities of the GCI are sufficiently close to the target value. Finally, two real examples are presented to illustrate the implementation of the proposed method. [ABSTRACT FROM AUTHOR]

Thechart has been known as a powerful tool to monitor the variability of the normal process. This paper considers the variable sampling interval (VSI)chart to improve the performance of the originalchart. A VSIchart based on the average run length (ARL) unbiased and average time to signal (ATS) unbiased criteria is developed when the in-control variance is known. The performance of the proposed chart is investigated in terms of the mean and standard deviation of the RL and total time to signal. The impact of parameter estimation on the proposed VSIchart is also evaluated. The results show that the performance of the control chart is seriously different from the known parameter case when the parameter is estimated. Then, a new VSIchart with the desired in-control ARL and ATS is considered when the in-control variance is estimated. In evaluating the usefulness of the VSI feature, the VSIcharts are compared with the fixed sampling interval (FSI) versions in terms of their statistical properties. The comparison results show that the VSIchart can detect the shifts in the process variance quicker than the corresponding FSI chart, regardless of whether the in-control variance is known or unknown. Finally, two examples are used to illustrate the proposed control chart. [ABSTRACT FROM AUTHOR]

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknownin-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible. [ABSTRACT FROM AUTHOR]

In this paper, we consider the conditional performance of the equal-tailed and average run lengths (ARL)-unbiased two-sided S2 charts when the in-control variance of a normal process is estimated. We derive the exact probability distributions of the conditional ARL for the two S2 charts. Then we evaluate the performance of each S2 chart in terms of the percentiles, mean and standard deviation of the conditional in-control ARL distribution. Because the parameter estimation seriously affects the conditional performance of these S2 charts, we propose an exact method to design the equal-tailed and ARL-unbiased S2 charts with desired conditional in-control performance. The results indicate that the new ARL-unbiased S2 chart has far smaller standard deviation ARL values and the unconditional ARL values are more close to the desired value than the corresponding new equal-tailed S2 chart. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The S2 chart has been known as a powerful tool to monitor the variability of the normal process. When the variance of the process is unknown, it needs to be estimated by Phase I samples. It is well known that there are serious effects of parameter estimation on the performance of the S2 chart based on known parameter assumption. If the effects of parameter estimation are not considered, it can lead to an increase in the number of false alarms and a reduction in the ability of the chart to detect process changes except for very small shifts in the variance. Based on the criterion of average run length (ARL) unbiased, a S2 control chart is developed when the in-control variance is estimated. The performance of the proposed control chart is also evaluated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control chart. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a new statistic is proposed to monitor the Weibull shape parameter when the sample is type II censored. The one-sided and two-sided average run length-unbiased control charts are derived based on the new monitoring statistic. The control limits of the proposed control charts depend on the sample size, the failure number and the false alarm rate. Using Monte Carlo simulation, the performance of the proposed control charts is studied and compared with the range-based charts proposed by Pascual and Li (2012), which is equivalent to the proposed control charts when r = 2. The simulation results show that the proposed control charts perform better than the ones of Pascual and Li (2012). This paper also evaluates the effects of parameter estimation on the proposed control charts. Finally, an example is used to illustrate the proposed control charts. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]