Product defect rate confidence bound with attribute and variable data

The upper confidence bound for a product defect rate is a very important index for evaluating the production process in industry. In this paper, we provide a bootstrap methodology to construct a (1?α)100% upper confidence bound for the overall defect rate of a product whose quality assessment involves multiple pass/fail binary data and multiple continuous data. When only the pass/fail data are included we propose using a bootstrap method which is consistent with the Clopper–Pearson one‐sided confidence interval. When only the continuous data are included the BC_a bootstrap method is recommended. These two methods are combined to provide an upper confidence bound for the overall defect rate of the product when multiple pass/fail binary data and multiple continuous data are present. All methods are clearly stated in algorithmic form, investigated through simulation and demonstrated using example data sets. In the simulation studies and examples the proposed algorithms show great advantages in both coverage probability and computational efficiency. Copyright © 2010 John Wiley & Sons, Ltd.     

An exact method for designing Shewhart and S2 control charts to guarantee in-control performance

The in-control performance of Shewhart and S² control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate that an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In this article we present simple formulas using the assumption of normality to compute the control limits and therefore, users do not have to use the bootstrap method. The advantage of our proposed method is in its simplicity for users; additionally, the control chart constants do not depend on the Phase I sample data.     

Nonlinear logistic regression mixture experiment modeling for binary data using dimensionally reduced components

This article motivates, presents, and illustrates an approach using nonlinear logistic regression (NLR) for modeling binary response data from a mixture experiment when the components can be partitioned into groups used to form dimensionally reduced components (DRCs). A DRC is formed from a linear combination of the components in a group having similar roles and/or effects of the same sign, where the linear combinations over all groups are normalized so that the DRC proportions sum to one. Linear combinations of a particular form provide for quantifying the effects of the remaining components in a group relative to a chosen component. This reason, plus dimensional reduction, are the primary motivations for the proposed DRC mixture experiment modeling approach. NLR is required because models expressed in terms of the DRCs are nonlinear in the parameters that specify the linear combinations. A method for obtaining nonparametric tolerance limits on the probability of a “success” for the binary response variable using a bootstrap approach is also presented. Finally, the article shows how DRCs provide for visualizing data and modeling results that otherwise would be impossible. A real database on the presence or absence of nepheline crystals in simulated nuclear waste glass is used to illustrate the NLR modeling and nonparametric tolerance limit approaches. The methodology is general and can be applied to other applications.     

Bayesian tolerance interval control limits for attributes

The probability content of standard control limits for attributes can vary because distribution parameters that appear in the control limits are estimated based on previous data. This paper proposes using Bayesian tolerance interval control limits which control the probability content at a specified level with a given confidence. Bayesian tolerance interval control limits are developed for np, p, c and u charts and are illustrated with four examples from the literature. Moreover, Bayesian tolerance interval control limits can be used for processes at start‐up. Published in 2002 by John Wiley & Sons, Ltd.     

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.     

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.     

New Methods for One-Sided Tolerance Limits for a One-Way Balanced Random-Effects ANOVA Model

The problem of determining one-sided tolerance limits or, equivalently, one-sided confidence limits on a quantile of a normal population with variance σ² = σ² _b + σ² _b is considered in this article. The tolerance limits are obtained using data from a one-way balanced randomeffects sample from this population. Two procedures are introduced—a simple approximate method based on an asymptotic expansion and a computer-intensive procedure that provides very nearly the nominal confidence in the presence of the nuisance variance-component ratio. For the computer-intensive procedure, tables are provided that eliminate the need for most of the computation for some important situations. Tolerance limits for strength are routinely used in structural design. Material strength, particularly for composite materials, often exhibits considerable between-batch variability. A tolerance limit that provides nearly the nominal confidence is usually preferable to a conservative limit, since a conservative tolerance limit underestimates the capability of the material. An aircraft-industry application to determining tolerance limits for composite material properties in the presence of between-batch variability is discussed.     

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.     

On Simultaneous Confidence Intervals for Multinomial Proportions

In this article we present a method for obtaining simultaneous confidence intervals for the parameters of a multinomial distribution, and we compare this method with the one suggested recently by Quesenberry and Hurst (1964). For the usual probability levels, we find, for example, that the confidence intervals introduced here have the desirable property that they are shorter than the corresponding intervals obtained by the Quesenberry-Hurst method. We also present methods for obtaining simultaneous confidence intervals for the differences among the parameters of the multinomial distribution, and we compare these methods with the one suggested earlier by Gold (1963) for studying linear functions of the multinomial parameters. For the usual probability levels, we find that the confidence intervals introduced in the present article have the desirable property that they are shorter than the corresponding intervals obtained by the Gold method applied to the differences among the multinomial parameters. In addition, we show how the methods presented here for studying the differences among the multinomial parameters can be modified in order to obtain simultaneous confidence intervals for the relative differences among the multinomial parameters.     

Bootstrapping extremes of random variables under power normalization

This paper deals with the asymptotic of bootstrap for the distribution of extremes under power normalization when the underlying distribution belongs to the domain of attraction of an extreme value distribution. We obtained the inconsistency, weak consistency and strong consistency of bootstrapping with appropriate choice of resample size when the normalizing constants are known. The same problem are investigated when the normalizing constants are unknown. The bootstrap for the joint distributions and the confidence intervals for the upper end of the distribution functionF are obtained.     

Reliability Analysis of Zero‐Failure Data with Poor Information

For costly and dangerous experiments, growing attention has been paid to the problem of the reliability analysis of zero‐failure data, with many new findings in world countries, especially in China. The existing reliability theory relies on the known lifetime distribution, such as the Weibull distribution and the gamma distribution. Thus, it is ineffective if the lifetime probability distribution is unknown. For this end, this article proposes the grey bootstrap method in the information poor theory for the reliability analysis of zero‐failure data under the condition of a known or unknown probability distribution of lifetime. The grey bootstrap method is able to generate many simulated zero‐failure data with the help of few zero‐failure data and to estimate the lifetime probability distribution by means of an empirical failure probability function defined in this article. The experimental investigation presents that the grey bootstrap method is effective in the reliability analysis only with the few zero‐failure data and without any prior information of the lifetime probability distribution. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of Nonparametric Binary Regression to Evaluate the Sensitivity of Explosives

The performance of slapper detonators depends on the sensitivity of the explosive material within the detonators to a voltage stimulus. A series of experiments is typically used to estimate the sensitivity of the explosive material by subjecting individual detonators to various stimulus levels. The traditional methods for selecting stimulus levels and analyzing the resulting binary data are based on a parametric probit model that relates the probability of detonation to the level of the voltage stimulus. In this article, we explore an alternative method for analyzing the binary data using nonparametric regression. The use of this alternative method is illustrated by analyzing the ungrouped binary data resulting from a series of 25 tests that were used to characterize a single lot of detonators. We conclude that nonparametric regression can be very useful in visualizing the underlying relationship between a binary response and a covariate, even when the sample size is relatively small.     

The influence of some blasting techniques on the probability of ignition of firedamp by permissible explosives

The ignition of firedamp by permissible explosives is assessed by means of gallery testing conducted by the Bruceton up-and-down method. Six test series were made in order to analyze the influence of several blasting practices in long-holes coal blasting, namely: use of slotted PVC pipes, detonating cord, salt cartridges and double (top and bottom) initiation. The parameters of the distributions of the probability of ignition are determined by the maximum likelihood method; normal, logistic, lognormal and Weibull distributions have been used. Confidence bands for the probability points are obtained both from the asymptotic standard errors of the parameters and by a bootstrap-like technique. The four distributions used give similar results in a rather ample probability range; discrepancies in the probability points are within 2% and in the confidence limits within 10% in a range of probability [0.1, 0.9] in most of the cases. The use of detonating cord is found to affect significantly the probability of ignition; the double initiation does also have an influence though not statistically significant at a 95% level; the use of salt cartridges, in the amount tested, has little effect in the ignition probability; the use of PVC pipe shows no effect.     

Discussion of “Estimation of Parameters for a Mixture of Normal Distributions” by Victor Hasselblad

Applications are outlined of the noncentral t-distribution to tolerance limits, to variables sampling plans, to confidence limits on a quantile, to confidence limits on a proportion, to the distribution of the sample coefficient of variation, and to the power of Student's t-test. The basic assumption is that there is available a random sample from a normal distribution with mean and variance unknown. Some of the mathematical properties of the noncentral t-distribution are also outlined. An extensive bibliography has been prepared and cross-referenced to several review journals and books. The bibliography covers references to tolerance limits and sampling plans based on the normal distribution whether or not the noncentral t-distribution is directly involved. Some selected references to Student's t-distribution are also included.     

Obtaining confidence intervals for Cpk using percentiles of the distribution of Ĉp

?_pk is used as an estimate of process capability and can reflect performance degradation due to both shifts in the process mean and variability. Exact upper and lower confidence limits for the actual parameter value are elusive. Objections to existing estimators focus on two areas: the difficulty of executing the needed computations and the excessive widths of those confidence bounds. The CPE estimator (so called since it relies on ?_p ) which we derive in this paper is formed by simply multiplying ?_pk by a value from the inverse chi‐square distribution. A PASCAL computer program is supplied which generates those percentiles. Our CPE is compared via simulation to other estimators and is shown to be the preferred selection in most cases by providing the requisite probability coverage and narrow interval widths. Copyright © 2001 John Wiley & Sons, Ltd.     

A note on the first-passage problem and VanMarcke’s approximation — short communication

Given a scalar, stationary, Markov process, this short communication presents a closed-form solution for the first-passage problem for a fixed threshold b. The derivation is based on binary processes and the general formula of Siegert [Siegert AJF. On the first-passage time probability problem. Physical Review 1951; 81:617–23]. The relation for the probability density function of the first-passage time is identical to the commonly used formula that was derived by VanMarcke [VanMarcke E. On the distribution of the first-passage time for normal stationary random processes. Journal of Applied Mechanics ASME 1975; 42:215–20] for Gaussian processes. The present derivation is based on more general conditions and reveals the criteria for the validity of the approximation. Properties of binary processes are also used to derive a hierarchy of upper bounds for any scalar process.     

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.     

The optimal schedule to produce a given number of acceptable parts with a specified confidence level

This paper deals with the problem of finding the optimal schedule for producing, with a probability α, a finite number H, of parts which have a diameter within specified tolerance limits. It is assumed that the diameter is a normally distributed variable that exhibits a linear trend in the process mean. The solution involves determining the optimal run size(s), as well as the specific number of runs of each size, required to produce at least H parts, with probability α, at minimum cost. A solution algorithm is provided and computational experience reported.     

A simultaneous confidence corridor for varying coefficient regression with sparse functional data

We consider a varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time-independent covariates with coefficients as functions of time-dependent covariates. Based on spline smoothing, we propose data-driven simultaneous confidence corridors for the coefficient functions with asymptotically correct confidence level. Such confidence corridors are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypotheses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p values on the effects of smoking, pre-infection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment.     

Inference on the Gamma Distribution

This study develops inferential procedures for a gamma distribution. Based on the Cornish–Fisher expansion and pivoting the cumulative distribution function, an approximate confidence interval for the gamma shape parameter is derived. The generalized confidence intervals for the rate parameter and other quantities such as mean are explored. The proposed generalized inferential procedures are extended to construct prediction limits for a single future measurement and for at least p of m measurements at each of r locations. The performance of the proposed procedures is evaluated using Monte Carlo simulation. The simulation results show that the proposed procedures are very satisfactory. Finally, three real examples are used to illustrate the proposed procedures. Supplementary materials for this article are available online.