A weighted variance capability index for general non‐normal processes

Process capability indices are considered to be one of the important quality measurement tools for the continuous improvement of quality and productivity. The most commonly used indices assume that process data are normally distributed. However, many studies have pointed out that the normally‐based indices are very sensitive to non‐normal processes. Therefore we propose a new process capability index applying the weighted variance control charting method for non‐normal processes to improve the measurement of process performance when the process data are non‐normally distributed. The main idea of the weighted variance method is to divide a skewed or asymmetric distribution into two normal distributions from its mean to create two new distributions which have the same mean but different standard deviations. In this paper we provide an example, a distribution generated from the Johnson family of distributions, to demonstrate how the weighted variance‐based process capability indices perform in comparison with another two non‐normal methods, namely the Clements and Johnson–Kotz–Pearn methods. This example shows that the weighted variance‐based indices are more consistent than the other two methods in estimating process fallout for non‐normal processes. Copyright © 1999 John Wiley & Sons, Ltd.     

Measuring the Process Yield for Circular Profiles

Two indices C_p(circular) and C_pk(circular) based on the functional method have been proposed to measure the process capability of circular profiles. However, these two indices only provide potential capability and a lower bound on the process yield, respectively. In this paper, we develop a new yield index S_pk(circular) for circular profiles. This index provides an exact measure of process yield. The asymptotic normal distribution of the estimated index is derived. The statistical inferences such as hypothesis testing, confidence interval, and lower confidence interval can be easily constructed. A simulation study is conducted to assess the performance of the proposed method. The simulation results confirm that the estimates are close to the true value and the coverage rates of the confidence intervals are greater than the 95% lower limit of the stated nominal in most cases. One real data set is used to illustrate the applicability of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Computing process capability indices for non‐normal data: a review and comparative study

When the distribution of a process characteristic is non‐normal, C_p and C_pk calculated using conventional methods often lead to erroneous interpretation of the process's capability. Though various methods have been proposed for computing surrogate process capability indices (PCIs) under non‐normality, there is a lack of literature that covers a comprehensive evaluation and comparison of these methods. In particular, under mild and severe departures from normality, do these surrogate PCIs adequately capture process capability, and which is the best method(s) in reflecting the true capability under each of these circumstances? In this paper we review seven methods that are chosen for performance comparison in their ability to handle non‐normality in PCIs. For illustration purposes the comparison is done through simulating Weibull and lognormal data, and the results are presented using box plots. Simulation results show that the performance of a method is dependent on its capability to capture the tail behaviour of the underlying distributions. Finally we give a practitioner's guide that suggests applicable methods for each defined range of skewness and kurtosis under mild and severe departures from normality. Copyright © 1999 John Wiley & Sons, Ltd.     

A New Method for Constructing Confidence Intervals for the Index Cpm

In the statistical literature on the study of the capability of processes through the use of indices, C_pm appears to have been one of the most widely used capability indices and its estimation has attracted much interest. In this article, a new method for constructing approximate confidence intervals or lower confidence limits for this index is suggested. The method is based on an approximation of the non‐central chi‐square distribution, which was proposed by Pearson. Its coverage appears to be more satisfactory compared with that achieved by any of the two most widely used methods that were proposed by Boyles, in situations where one is interested in assessing a lower confidence limit for C_pm. This is supported by the results of an extensive simulation study. Copyright © 2004 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.     

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.     

Univariate and multivariate process yield indices based on location-scale family of distributions

Several measures of process yield, defined on univariate and multivariate normal process characteristics, have been introduced and studied by several authors. These measures supplement several well-known Process Capacity Indices (PCI) used widely in assessing the quality of products before being released into the marketplace. In this paper, we generalise these yield indices to the location-scale family of distributions which includes the normal distribution as one of its member. One of the key contributions of this paper is to demonstrate that under appropriate conditions, these indices converge in distribution to a normal distribution. Several numerical examples will be used to illustrate our procedures and show how they can be applied to perform statistical inferences on process capability.     

A Bayesian procedure for process capability assessment

The usual practice of judging process capability by evaluating point estimates of some process capability indices has a flaw that there is no assessment on the error distributions of these estimates. However, the distributions of these estimates are usually so complicated that it is very difficult to obtain good interval estimates. In this paper we adopt a Bayesian approach to obtain an interval estimation, particularly for the index C_pm. The posterior probability p that the process under investigation is capable is derived; then the credible interval, a Bayesian analogue of the classical confidence interval, can be obtained. We claim that the process is capable if all the points in the credible interval are greater than the pre‐specified capability level ω, say 1.33. To make this Bayesian procedure very easy for practitioners to implement on manufacturing floors, we tabulate the minimum values of Ĉ_pm/ω, for which the posterior probability p reaches the desirable level, say 95%. For the special cases where the process mean equals the target value for C_pm and equals the midpoint of the two specification limits for C_pk, the procedure is even simpler; only chi‐square tables are needed. Copyright © 1999 John Wiley & Sons, Ltd.     

Bootstrap Confidence Limits for Short-Run Capability Indices

The capability indices are widely used by quality professionals as an estimate of process capability. Recently, techniques and tables were developed to construct confidence limits for each index. These techniques are based on the assumption that the underlying process is normally distributed. In industries it may not be possible to get large samples, and, hence, the normality assumption is often violated. For the short-run production processes where sample size is small, appropriate indices can be developed. In this article, the capability indices such as C_p, C_pk, and C_pm are modified, and appropriate capability indices are constructed. A further bootstrap technique is used to define the confidence intervals. A simulation using two distributions (one normal and the other nonnormal) is conducted, and a comparison is made to show the performances of the three nonparametric confidence intervals.     

Comparing Two Process Capability Indices under Balanced One‐Way Random Effect Model

Process capability indices such as C_p, C_pk, C_pmk and C_pm are widely used in manufacturing industries to provide a quantitative measurement of the performance of the products. In this article, we derived generalized confidence intervals for the difference between process capability indices for two processes under one‐way random effect model. Our study provides coverage probability close to the nominal value in almost all cases as shown via simulation. An example from industrial contexts is given to illustrate the results. Copyright © 2012 John Wiley & Sons, Ltd.     

Re-evaluating the process capability indices for non-normal distributions

Classical process capability indices (PCIs) C _p , C _pu , C _pl and C _pk can indicate the potential process capability accurately when the quality characteristic of the product is normally distributed. When the process has a non-normal distribution, classical PCIs will be inappropriate and can misled the assessment of process capability. Zwick (1995), Schneider and Pruett (1995-96) and Tong and Chen (1998) proposed various PCIs for non-normal distributions. This paper compares the accuracy of these indices for several selected non-normal distributions based on the proportion of non-conformity of manufactured product. The results indicate that these PCIs lead to a larger number of errors in various combinations of shape parameters and specification limits. Therefore, this paper proposes three indices, S _pu , S _pl and S _pk , which can reflect accurately the proportion of nonconformity in either normal or non-normal distributions.     

An application of non-normal process capability indices

Numerous process capability indices, including C_p, C_pk, C_pm, and C_pmk, have been proposed to provide measures of process potential and performance. In this paper, we consider some generalizations of these four basic indices to cover non-normal distributions. The proposed generalizations are compared with the four basic indices. The results show that the proposed generalizations are more accurate than those basic indices and other generalizations in measuring process capability. We also consider an estimation method based on sample percentiles to calculate the proposed generalizations, and give an example to illustrate how we apply the proposed generalizations to actual data collected from the factory. © 1997 John Wiley & Sons, Ltd.     

Distributional Properties of Multivariate Process Capability Indices under Normal and Non‐normal Distributions

Multivariate capability analysis has been the focus of study in recent years, during which many authors have proposed different multivariate capability indices. In the operative context, capability indices are used as measures of the ability of the process to operate according to specifications. Because the numerical value of the index is used to conclude about the capability of the process, it is essential to bear in mind that almost always that value is obtained from a sample of process units. Therefore, it is really necessary to know the properties that the indices have when they are calculated on sampling information, in order to assess the goodness of the inferences made from them. In this work, we conduct a simulation study to investigate distributional properties of two existing indices: NMCpm index based on ratio of volumes and Mp₂ index based on principal component analysis. We analyze the relative bias and the mean square error of the estimators of the indices, and we also obtain their empirical distributions that are used to estimate the probability that the indices classify correctly a process as capable or as incapable. The results allow us to recommend the use of one of these indices, as it has shown better properties. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of Testing Normality on Estimating Process Capability Indices

Process capability indices are used to measure whether a manufacturing process meets the specifications. The studies of these indices are usually based on the assumption that the process follows a normal distribution. These include the indices C_p, C_pk, C_pm, and C_pmk. When the investigator is uncertain whether the process follows a normal distribution, a test of normality may be used to resolve the uncertainty. If the test accepts the null hypothesis that the process follows a normal distribution, the investigator uses C_p, C_pk, C_pm, or C_pmk. If the test rejects the null hypothesis, the investigator uses indices under non-normal distributions. Therefore the test of normality is a preliminary test that determines the form of the distribution and the index to use. In this paper we study the effect of the preliminary test of normality on the estimation of the four process capability indices mentioned above.     

Generalized confidence intervals for process capability indices

The concept of generalized confidence intervals is used to derive lower confidence limits for some of the commonly used process capability indices. For the cases where approximate lower confidence limits are already available, numerical comparisons are made among the available approximations and the generalized lower confidence limit. The numerical results indicate that the generalized confidence interval does provide coverage probabilities very close to the nominal confidence level. Two examples are given to illustrate the results. Copyright © 2006 John Wiley & Sons, Ltd.     

A class of generalized Birnbaum-Saunders distributions for reliability modeling

In this paper, we generalize the Birnbaum-Saunders (BS) distribution by two ways. One is based on the mixture representation of BS distribution, and a flexible weight is adopted to describe the kurtosis of the distribution. The other way is based on the transformation property of BS distribution, and we incorporate a power parameter in the transformation to describe the skewness of the distribution. Then a four-parameter BS distribution including skewness and kurtosis parameters is induced by combining the two ways. The properties of these generalized BS distributions are investigated. Then, the expectation maximization (EM) algorithm is proposed to estimate the parameters. Real data analysis is performed to illustrate the superiority of the generalized BS distributions. Finally, some potential generalizations are discussed.     

Statistical Analysis for Decision Making

In this article we consider a generalization of the univariate g-and-h distribution to the multivariate situation with the aim of providing a flexible family of multivariate distributions that incorporate skewness and kurtosis. The approach is to modify the underlying random variables and their quantiles, directly giving rise to a family of distributions in which the quantiles rather than the densities are the foci of attention. Using the ideas of multivariate quantiles, we show how to fit multivariate data to our multivariate g-and-h distribution. This provides a more flexible family than the skew-normal and skew-elliptical distributions when quantiles are of principal interest. Unlike those families, the distribution of quadratic forms from the multivariate g-and-h distribution depends on the underlying skewness. We illustrate our methods on Australian athletes data, as well as on some wind speed data from the northwest Pacific.     

A novel statistical tolerance analysis method for assembled parts

Tolerance analysis of an assembly is an important issue for mechanical design. Among various tolerance analysis methods, statistical analysis is the most commonly employed method. However, the conventional statistical tolerance method is often based on the normal distribution. It fails to predict the resultant tolerance of an assembly of parts with non-normal distributions. In this paper, a novel method based on statistical moments is proposed. Tolerance distributions of parts are first transferred into statistical moments that are then used for computing tolerance stack-up. The computed moments, particularly the variance, the skewness and the kurtosis, are then mapped back to probability distributions in order to calculate the resultant tolerance of the assembly. The proposed method can be used to analyse the resultant tolerance specification for non-normal distributions with different skewness and kurtosis. Simulated results showed that tail coefficients of different distributions with the same kurtosis are close to each other for normalised probabilities between ?3 and 3. That is, the tail coefficients of a statistical distribution can be predicted by the coefficients of skewness and kurtosis. Two examples are illustrated in the paper to demonstrate the proposed method. The predicted resultant tolerances of the two examples are only 0.5% and 1.5% differences compared with that by the Monte Carlo simulation for 1,000,000 samples. The proposed method is much faster in computation with higher accuracy than conventional statistical tolerance methods. The merit of the proposed method is that the computation is fast and comparatively accurate for both symmetrical and unsymmetrical distributions, particularly when the required probability is between ±2σ and ±3σ.     

Capability Testing Based on Cpm with Multiple Samples

Numerous process capability indices have been proposed in the manufacturing industry to provide unitless measures on process performance, which are effective tools for quality improvement and assurance. Most existing methods for capability testing are based on the distribution frequency approaches. Recently, Bayesian approaches have been proposed for testing capability indices C_p and C_pm but restricted to cases with one single sample. In this paper, we consider estimating and testing capability index C_pm based on multiple samples. We propose accordingly a Bayesian procedure for testing C_pm. Based on the Bayesian procedure, we develop a simple but practical procedure for practitioners to use in determining whether their manufacturing processes are capable of reproducing products satisfying the preset capability requirement. A process is capable if all the points in the credible interval are greater than the pre‐specified capability level. To make the proposed Bayesian approach practical for in‐plant applications, we tabulate the minimum values of for which the posterior probability p reaches various desirable confidence levels. Copyright © 2004 John Wiley & Sons, Ltd.     

An examination of the literature distributions of three scientific specialties

An examination was conducted of the distributions produced by historical treatments of three scientific specialties: quantum mechanics, plate tectonics, and endocrinology. A citation analysis approach was employed to generate a frequency distribution for year of publication of literature referenced by historians. The observed values were normalized and tested for goodness of fit to each other using a Pearson goodness of fit test. The results indicated that the three distributions were not equivalent. Other parameters of the three distributions did show similarities using aDunn planned comparison approach. The skewness of the three distributions was very similar and plate tectonics and endocrinology were similar in terms of kurtosis. The major conclusion reached was that there were major differences in the three distributions but some similarities in particular parameters were evident. Additional work is necessary to determine causal factors for the differences as well as similarities.