Bootstrap confidence intervals of process capability indices Cpy and CNpmk using different methods of estimation for Frechet distribution

In this paper, we considered the Bayesian estimators under reference and Jeffery's priors and maximum likelihood estimators to estimate the unknown parameters of the process capability indices Spmk, Spmkc, and Cs forFrechetdistribution. Further, we developed bootstrap confidence intervals for aforementioned process capability indices based on above‐mentioned estimators. Monte Carlo simulations are performed to investigate the performance of process capability indices through skewness, kurtosis, mean square error and widths of bootstrap confidence intervals for small, moderate, and large sample sizes. Simulations results indicate that the Bayesian estimator under reference prior outperforms even in small sample sizes, and all performed equally well for larger sample sizes. Moreover, the average width for bootstrap confidence interval for Cs is least in all. Finally, real data is analyzed for illustration purposes.

The process capability indices are important numerical measures in statistical quality control. Well-known process capability indices are constructed under the process distribution is normal. Unfortunately, this situation is rather not realistic. This paper focuses on the half logistic distribution. The bootstrap confidence intervals for the difference between two process capability indices for the mentioned distribution are proposed. The bootstrap confidence intervals considered in this paper consist of the standard bootstrap confidence interval, the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Simulation results showed that the estimated coverage probabilities of the standard bootstrap confidence interval get closer to the nominal confidence level than those of the other bootstrap confidence intervals.

Process capability index is an effective indicator for gauging the capability of a process potential and performance. It is used to quantify the relation between the actual performance of the process and the preset specifications of the product. In this article, we utilize bootstrap re-sampling simulation method to construct bootstrap confidence intervals, namely, standard bootstrap, percentile bootstrap, Student’s t bootstrap (STB) and bias-corrected percentile bootstrap to study the difference between two generalized process capability indices $$C_{\text {pTk1}}$$ and $$C_{\text {pTk2}}$$ ($$\delta =C_{\text {pTk1}}-C_{\text {pTk2}}$$) and to select the better of the two processes or manufacturer’s (or supplier’s) through simulation when the underlying distribution is Burr XII distribution. The model parameters are estimated by maximum likelihood method. The proposed four bootstrap confidence intervals can be effectively employed to determine which one of the two processes or manufacturers (or suppliers) has a better process capability. Monte Carlo simulations are performed to compare the performances of the proposed bootstrap confidence intervals for $$\delta $$ in terms of their estimated average widths, coverage probabilities and relative coverages. Simulation results showed that the estimated average width and relative coverages of the STB confidence interval perform better than their counterparts. Finally, real data are presented to illustrate the bootstrap confidence intervals of the difference between two process capability indices.

A process capability index (PCI) meant for assessing the capability of the concerned manufacturing process to manufacturer’s products as per specifications pre-set by the product designers or customers. In this article, we utilize bootstrap re-sampling simulation method to construct bootstrap confidence intervals, namely, standard bootstrap (s-boot), percentile bootstrap (p-boot), and bias-corrected percentile bootstrap (BCp-boot) for the difference between two indices ( $$C_\mathrm{pyk1}-C_\mathrm{pyk2}$$ ) through simulation when the underlying distribution is inverse Lindley distribution. Maximum-likelihood method is used to estimate the parameter of the model. The proposed bootstrap confidence intervals can be effectively employed to determine which one of the two processes or manufacturer’s (or supplier’s) has a better process capability. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals of ( $$C_\mathrm{pyk1}-C_\mathrm{pyk2}$$ ). Simulation results showed that the estimated coverage probabilities of the standard bootstrap confidence interval perform better than their counterparts. Finally, a simulated data and a real data are presented to illustrate the bootstrap confidence intervals of the difference between two PCIs.

In this article, non-informative priors are investigated for a zero-inflated Poisson distribution with two parameters: the probability of zeros and the mean of the Poisson part. Both the reference prior and the Jeffreys prior are derived and shown to be second-order matching priors when only the mean of the Poisson part is of interest. However, when the probability of zeros is of interest, the reference prior is still a second-order matching prior, whereas the Jeffreys prior is not so. Furthermore, when both parameters are of interest, the reference prior is a unique second-order matching prior. Frequentist coverage probabilities of the posterior confidence sets based on the Jeffreys and reference priors are compared with each other using Monte Carlo simulations and with confidence sets based on the maximum likelihood estimation.

ABSTRACTOne of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.

This study aims to develop and compare the maximum likelihood, least square, weighted least square, Anderson- Darling, right-tail Anderson-Darling, Cramér-von Mises, maximum of product spacings, and Bayesian estimators for the Marshall-Olkin inverse log-logistic distribution. The Bayesian estimators are developed based on the reference and Jeffreys priors. Since closed-form Bayesian estimators cannot be obtained, an approximate Bayesian approach is proposed using the idea of Laplace’s approximation, and results are compared via Monte Carlo simulations. Additionally, three real-life data sets are analyzed for illustration. The study shows that the Bayesian estimators based on reference and Jeffreys priors outperform their counterparts in terms of mean squared errors.

Process capability analysis (PCA) is a tool for measuring a process’s ability to meet specification limits (SLs), which the customers define. Process capability indices (PCIs) are used for establishing a relationship between SLs and the considered process’s ability to meet these limits as an index. PCA compares the output of a process with the SLs through these capability indices. If the customers’ needs contain vague or imprecise terms, the classical methods are inadequate to solve the problem. In such cases, the information can be processed by the fuzzy set theory. Recently, ordinary fuzzy sets have been extended to several new types of fuzzy sets such as intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and spherical fuzzy sets. In this paper, a new extension of intuitionistic fuzzy sets, which is called penthagorean fuzzy sets, is proposed, and penthagorean fuzzy PCIs are developed. The design of production processes for COVID-19 has gained tremendous importance today. Surgical mask production and design have been chosen as the application area of the penthagorean fuzzy PCIs developed in this paper. PCA of the two machines used in surgical mask production has been handled under the penthagorean fuzzy environment.

A note on noninformative priors for Weibull distributions

This paper deals with construction of confidence intervals for process capability index using bootstrap method (proposed by Chen and Pearn in Qual Reliab Eng Int 13(6):355–360, 1997) by applying simulation technique. It is assumed that the quality characteristic follows type-II generalized log-logistic distribution introduced by Rosaiah et al. in Int J Agric Stat Sci 4(2):283–292, (2008). Discussed different bootstrap confidence intervals for process capability index. Maximum likelihood method is considered for obtaining the estimators of the parameter. Monte Carlo simulation technique is applied to find out the coverage probabilities and average widths of the bootstrap confidence intervals. The results are illustrated with real data sets.

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.

For enhancing the quality and productivity, the use of process capability index (PCI) has become significant in statistical process control, designed to quantify the relation between the actual performance of the process and its specified requirements. Confidence interval is an important part of PCI because it is an estimated value and provides much more information about the population characteristic of interest than does a point estimate. In this article, bootstrap confidence intervals of non-normal PCI , is studied through simulation when the underlying distribution is exponential power distribution. Maximum likelihood method is used to estimate the parameters of the model. Four (parametric as well as non-parametric) bootstrap confidence intervals, namely standard bootstrap (s-boot), percentile bootstrap (p-boot), Student’s t bootstrap (t-boot), and bias-corrected percentile bootstrap (-boot), are considered for obtaining confidence intervals of . A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Simulation results showed that among these (s-boot, p-boot, t-boot, and -boot) confidence intervals, the performances of the s-boot confidence intervals is the best in terms of overage probabilities. Finally, three real data-sets are analyzed for illustrative purposes.

Confidence intervals for process capability index using bootstrap method (Chen and Pearn, Qual Reliab Eng Int 13(6), 355–360, 1997) are constructed through simulation assuming that the underlying distribution is exponentiated Frechet distribution (EFD). Parameters are estimated by Maximum likelihood (ML) method. Also obtain the estimated coverage probabilities and average widths of the bootstrap confidence intervals through Monte Carlo simulation. Illustrate the process capability indices for EFD using some numerical examples.

In this article, we propose a new process capability index called C pmc ′ which is based on asymmetric loss function (linear exponential) and tolerance cost function for a normal process which provides a tailored way of incorporating the loss and tolerance cost in capability analysis. Next, we estimate the proposed PCI C pmc ′ when the process follows the normal distribution using six classical methods of estimation and we compare the performance of the considered methods of estimation in terms of their mean squared errors through simulation study. Besides, five bootstrap methods are employed for constructing the confidence intervals for the index C pmc ′ . The performance of the bootstrap confidence intervals (BCIs) are compared in terms of average width and coverage probabilities using Monte Carlo simulation. Finally, for illustrating the effectiveness of the proposed index and methods of estimation and BCIs, two real data sets from electronic industries are re-analyzed.

In this paper, a set of important objective priors are examined for the Bayesian estimation of the parameters present in the Poisson-Exponential distribution PE. We derived the multivariate Jeffreys prior and the Maximal Data Information Prior. Reference prior and others priors proposed in the literature are also analyzed. We show that the posterior densities resulting from these approaches are proper although the respective priors are improper. Monte Carlo simulations are used to compare the efficiencies and to assess the sensitivity of the choice of the priors, mainly for small sample sizes. This simulation study shows that the mean square error, mean bias and coverage probability of credible intervals under Gamma, Jeffreys' rule and Box & Tiao priors presented equal results, whereas Jeffreys and Reference priors showed the best results. The MDIP prior had a worse performance in all analyzed situations showing not to be indicated for Bayesian analysis of the PE distribution. A real data set is analyzed for illustrative purpose of the Bayesian approaches.