On the bootstrap confidence intervals of the process incapability index Cpp

The process incapability index C_pp is an indicator, introduced by Greenwich and Jahr-Schaffrath, for evaluating the capability of a process. When C_pp is applied to evaluate a process, estimating the confidence interval of C_pp is important for statistical inference on the process. Calculating the confidence interval for a process index usually needs the assumption about the underlying distribution. Bootstrapping is a non-parametric, but computer intensive, estimation method. In the present paper we report the results of a simulation study on the behavior of four 95% bootstrap confidence intervals (i.e. standard bootstrap, percentile bootstrap, biased-corrected percentile bootstrap, and biased-corrected and accelerated bootstrap) for estimating C_pp when data are from a specific Burr distribution, which is used to represent various probability distributions. A detailed discussion of the simulation results is presented and some conclusions are provided.     

Interval Estimation for the Smaller-the-Better Type of Signal-to-Noise Ratio using Bootstrap Method

The signal-to-noise ratio is an indicator, introduced by Taguchi, for evaluating the experimental data in robust design. Estimating the confidence interval of the signal-to-noise ratio is an important topic in data analysis of robust design. Calculating the confidence interval for a parameter usually needs the assumption about the underlying distributions. Bootstrapping is a nonparametric, but computer-intensive estimation method. In this article, we present the results of a simulation study on the behavior of three 95% bootstrap confidence intervals (i.e., SB, PB, and BCPB) for estimating the smaller-the-better signal-to-noise ratio when the data are from either a normal distribution or one of the Burr distributions. A detailed discussion of the simulation results is presented and some recommendations are given.     

A graphical approach to obtaining confidence limits of Cpk

The process capability index C_pk has been widely used as a process performance measure. In practice this index is estimated using sample data. Hence it is of great interest to obtain confidence limits for the actual index given a sample estimate. In this paper we depict graphically the relationship between process potential index (C_p), process shift index (k) and percentage non-conforming (p). Based on the monotone properties of the relationship, we derive two-sided confidence limits for k and C_pk under two different scenarios. These two limits are combined using the Bonferroni inequality to generate a third type of confidence limit. The performance of these limits of C_pk in terms of their coverage probability and average width is evaluated by simulation. The most suitable type of confidence limit for each specific range of k is then determined. The usage of these confidence limits is illustrated via examples. Finally a performance comparison is done between the proposed confidence limits and three non-parametric bootstrap confidence limits. The results show that the proposed method consistently gives the smallest width and yet provides the intended coverage probability. © 1997 John Wiley & Sons, Ltd.     

An Empirical Bayes Process Monitoring Technique for Polytomous Data

When a product item is tested, usually one has more information than just pass or fail. Often there are categories of failure modes. The purpose of this paper is to develop a method to monitor the fractions of the tested items falling into different categories of pass/fail modes. Using the multinomial model with Dirichlet prior, we describe the theory underlying an empirical Bayes approach to monitoring polytomous data generated in manufacturing processes. A pseudo maximum likelihood estimator (PMLE) and the method‐of‐moments estimator (MME) of the hyperparameters of the prior distribution are considered and compared by a simulation study. It is found that the PMLE performs slightly better than the MME. A monitoring scheme based on the marginal distributions of the observed pass/fail fractions is proposed. The average run length behavior of the proposed monitoring scheme is investigated. Finally, an example to illustrate the use of the technique is given. Copyright © 2004 John Wiley & Sons, Ltd.     

Discussion: Fuzzy Thinking

This article investigates computation of pointwise and simultaneous tolerance limits under the logistic regression model for binary data. The data consist of n binary responses, where the probability of a positive response depends on covariates via the logistic regression function. Upper tolerance limits are constructed for the number of positive responses in m future trials for fixed as well as varying levels of the covariates. The former provides pointwise upper tolerance limits, and the latter provides simultaneous upper tolerance limits. The upper tolerance limits are obtained from upper confidence limits for the probability of a positive response, modeled using the logistic function. To compute pointwise upper confidence limits for the logistic function, likelihood-based asymptotic methods, small sample asymptotics, as well as bootstrap methods are investigated and numerically compared. To compute simultaneous upper tolerance limits, a bootstrap approach is investigated. The problems have been motivated by an application of interest to the U.S. Army, dealing with the testing of ballistic armor plates for protecting soldiers from projectiles and shrapnel, where the success probability depends on covariates such as the projectile velocity, size of the armor plate, etc. Such an application is used to illustrate the tolerance interval computations in the article. We provide the R codes used for the calculations presented in the examples in the article as supplementary material, available online.     

The Generalized Likelihood Ratio Chart for Monitoring a Proportion with Autocorrelation

When monitoring a proportion p, it is usually assumed that the binary observations are independent. This paper investigates the problem of monitoring p when the binary observations follow a two‐state Markov chain model with first‐order dependence. A Markov binary generalized likelihood ratio (MBGLR) chart based on a likelihood ratio statistic with an upper bound on the estimate of p is proposed. The MBGLR chart is used to monitor a continuous stream of autocorrelated binary observation. The MBGLR chart with a relatively large upper bound has good overall performance over a wide range of shifts. The extra number of defectives is defined to measure the loss when using control charts for monitoring p. The MBGLR chart is optimized over a range of upper bounds for the MLE of p. The numerical results show that the optimized MBGLR chart has a smaller extra number of defectives than the optimized Markov binary cumulative sum chart that can detect a shift in p much faster than a Shewhart‐type chart. Copyright © 2014 John Wiley & Sons, Ltd.     

An improved approach for process performance evaluation with the consideration of process yield and quality loss

In recent years, several process capability indices have received considerable research attention and increased usage in evaluating process performance and purchasing decisions in manufacturing industries. In particular, the third-generation capability index C_pmk is constructed by taking process yield and quality loss into consideration. Unfortunately, the sampling distribution of the estimated C_pmk is complicated, making the establishment of the exact confidence interval very difficult. Thus, this study applies the concept of the generalised pivotal quantity (GPQ) to derive the generalised confidence interval (GCI) for the index C_pmk. To examine the performance and the effectiveness of the GCI approach, a series of simulations is undertaken. Three types of bootstrap methods are also implemented and compared with the proposed GCI approach. The results reveal that the proposed GCI approach is superior to the three bootstrap methods since the obtained coverage rates are very close to the nominal confidence level in most of our studied cases even for small sample sizes. Therefore, this article recommends the GCI approach for evaluating process performance in real applications.     

The Advantages of Continuous Measurements over Pass/Fail Data

This article provides a justification for continuous measurements being more informative than pass/fail data. Namely, more information is provided for the same sample size or a smaller sample size is required for the same information. In the article, inference of the conformance probability, the probability of measurements meeting specifications, is considered. It is shown that continuous measurements provide dramatic advantages, especially when the conformance probability is high.     

Confidence limits for compliance testing using mixed acceptance criteria

For manufactured items sold by weight or volume, this article considers mixed acceptance criteria that put limits on the sample mean or on an upper confidence limit based on the sample mean and on the number of individual sample units that are nonconforming. For a normally distributed quality characteristic of interest, this article develops lower confidence limits for the mixed acceptance criteria applying the concept of a generalized pivotal quantity and applying a bias-corrected and accelerated parametric bootstrap. The accuracy of the confidence limits is assessed using estimated coverage probabilities, and the results are illustrated with an example.     

Process Yield Analysis for Linear Within‐Profile Autocorrelation

In many industrial applications, the quality of a process or product can be characterized by a function or profile. Owing to spatial autocorrelation or time collapse, the assumption of the observations within each profile that are uncorrelated is violated. This paper aims at evaluating the process yield for linear within‐profile autocorrelation. We present an approximate lower confidence bound for S_pkA when the observations within each profile follow a first‐order autoregressive AR(1) model. A simulation study is conducted to assess the performance of the proposed method. The simulation results confirm that the proposed method performs well for the bias, the standard deviation, and the coverage rate. One real example is used to demonstrate the applications of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Worst case product performance verification with electromagnetic interference test applications

When many variables are prevalent, it is difficult systematically to identify the optimum configuration for product certification. This need can occur in areas such as testing performance of closed-loop servo systems and testing products for electromagnetic interference (EMI). This paper proposes a test strategy to identify a 'determined worst case expected' product configuration and a criterion for pass/fail. The process was developed originally for EMI test applications; however, it can also be useful for other product performance specifications.     

Approximate Confidence Intervals for the Log‐Normal Standard Deviation

In this work, approximate confidence intervals are derived for the standard deviation of a log‐normal distribution. Simulations are conducted to evaluate the coverage probability, average length, and coverage bias of the derived approximate confidence intervals. The simulation results indicate that the proposed approximate confidence intervals perform reasonably well when the standard deviation of the log‐transformed variable is small. The approximate confidence intervals are applied to a phase I pharmacokinetic study and a real data set concerning a lab testing of reference engine oil. Copyright © 2015 John Wiley & Sons, Ltd.     

Interval estimation for inverse Gaussian distribution

In this paper, we consider interval estimation for the inverse Gaussian (IG) distribution. Using generalized pivotal quantity method, we derive the generalized confidence intervals (GCIs) for the model parameters and some quantities such as the quantile, the reliability function of the lifetime, the failure rate function, and the mean residual lifetime. We verify that the GCI of the scale parameter is the same as its commonly used exact CI. We also obtain the generalized prediction intervals (GPIs) for future failure times based on the observed failure data set. In addition, we get the GCI for the reliability of the stress–strength model when the stress and strength variables follow the IG distributions with different parameters. We compare the proposed GCIs and GPIs with the Wald CIs and bootstrap-$p$ CIs by simulation. The simulation results show that the proposed GCIs and GPIs are superior to the Wald CIs and the bootstrap-p CIs in terms of the coverage probability. Finally, two examples are used to illustrate the proposed procedures.     

Yield‐Related Process Capability Indices for Processes of Multiple Quality Characteristics

Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate C_pk and C_p (denoted by BC_pk and BC_p, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MC_pk and MC_p for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MC_pk. An algorithm suitable for such processes is developed to compute the natural estimate of MC_pk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MC_pk, a measure often used by producers for quality assurance to consumers. As for BC_p, we first modify the original definition with a simple preprocessing step to make BC_p scale‐invariant. A very efficient algorithm is developed for computing a natural estimator of BC_p. This new approach of BC_p can be easily extended to MC_p for multivariate processes. For BC_p, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright © 2012 John Wiley & Sons, Ltd.     

Matrices With Applications in Statistics

Sample size determination methods are considered for hypothesis testing about location parameters of two continuous distributions using independent random samples and the Mann–Whitney–Wilcoxon (MWW) test, for a specified shift, size, and power. These methods are based on the so-called “Noether's formula,” derived from a normal approximation to the power of the MWW test, and a pilot sample from each of the distributions. Compared with alternate methods, including the one using bootstrap similar to that of Hamilton and Collings (1991) the new methods are shown to be at least as good in terms of accuracy and substantially more efficient in terms of speed and variability. The simpler method, using linearly smoothed empirical cdf's of the pilot samples and requiring no bootstrapping, is recommended for practical use. Extensions and adaptations for obtaining an upper confidence bound on the sample size and to general linear rank tests are indicated.     

A Remark on Multiple Comparison Methods

Quantitative measurement is an accepted ideal, but pass–fail inspection remains a fact of life, even in high-technology industries. For pass–fail data, variance components do not separate gauge and material variation. This article focuses on maximum likelihood estimation of conditional misclassification rates, with and without reference evaluations to anchor the analysis. Likelihood-based confidence intervals and testing for reproducibility effects are also discussed.     

Confidence interval for the mean of non‐normal data

The problem of constructing a confidence interval for the mean of non‐normal data is considered. The Bootstrap method and the Box–Cox transformation method of constructing the confidence interval are compared with the normal theory method. Simulation studies are used to evaluate the performance of these different methods of constructing confidence intervals. The result is not surprising; the Bootstrap method is more effective and efficient than the Box–Cox transformation method and the normal theory method in this simulation study. A real example demonstrates the ability of these methods to construct a confidence interval for the mean of audit accounting data. Copyright © 2001 John Wiley & Sons, Ltd.     

Classical and Bayesian inference of Cpy for generalized Lindley distributed quality characteristic

The process capability index (PCI) is a quality control–related statistic mostly used in the manufacturing industry, which is used to assess the capability of some monitored process. It is of great significance to quality control engineers as it quantifies the relation between the actual performance of the process and the preset specifications of the product. Most of the traditional PCIs performed well when process follows the normal behaviour. However, using these traditional indices to evaluate a non‐normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, C_py, suggested by Maiti et al, which can be used for normal as well as non‐normal random variables. This article addresses the different methods of estimation of the PCI C_py from both frequentist and Bayesian view points of generalized Lindley distribution suggested by Nadarajah et al. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, least square and weighted least square estimators, and maximum product of spacings estimators. Next, we consider Bayes estimation under squared error loss function using gamma priors for both shape and scale parameters for the considered model. We use Tierney and Kadane's method as well as Markov Chain Monte Carlo procedure to compute approximate Bayes estimates. Besides, two parametric bootstrap confidence intervals using frequentist approaches are provided to compare with highest posterior density credible intervals. Furthermore, Monte Carlo simulation study has been carried out to compare the performances of the classical and the Bayes estimates of C_py in terms of mean squared errors along with the average width and coverage probabilities. Finally, two real data sets have been analysed for illustrative purposes.     

New Developments in the Applications of Bayesian Methods

This article presents an upper confidence bound for a measure of reliability/maintainability of a series system in which component repair is assumed to restore the system to its original status. Under exponentiality of component-repair time and survival time distributions, the measure M is the ratio of expected system-repair time to expected system-survival time. Moreover, under this exponentiality assumption, the bound is uniformly minimum among all bounds of specified level that are nondecreasing functions of the maximum likelihood estimator of M. As a matter of practical interest, the measure M has been incorporated into Military Standard MIL-STD-470B and is an official statistic of the Air Force Reliability Maintainability Information System.