Generalized Confidence Interval Research Articles

Assessing high‐quality process performance using the quality‐yield index: An innovative methodology

AbstractManufacturers must meet high‐quality standards and exceed customer expectations to stay competitive due to significant technological advancements in recent decades. While implementing the yield measure is useful for achieving process performance by focusing on products that fall within specified limits, it does not accommodate specific customer requirements, particularly when a product's quality characteristic deviates from target value. To address this need, the quality‐yield index (Q‐yield) has been proposed, which combines the process‐yield index and loss‐based capability index, providing a more advanced performance measure. However, the Q‐yield index's confidence interval is challenging to derive due to the complicated sampling distribution involved. Several existing methods have attempted to construct an approximate confidence interval but none have performed well. Therefore, this article proposes an innovative approach, called the generalized confidence intervals (GCIs), that utilizes the idea of generalized pivotal quantities to establish the confidence interval for the Q‐yield index. The proposed approach is evaluated through simulations and compared to existing methods. The results reveal that the proposed approach provides the most accurate results for constructing the lower confidence bound of the Q‐yield index. This approach is recommended to evaluate process performance using the Q‐yield index for high‐quality customer requirements.

Confidence intervals for functions of signal-to-noise ratio with application to economics and finance

PurposeConfidence intervals play a crucial role in economics and finance, providing a credible range of values for an unknown parameter along with a corresponding level of certainty. Their applications encompass economic forecasting, market research, financial forecasting, econometric analysis, policy analysis, financial reporting, investment decision-making, credit risk assessment and consumer confidence surveys. Signal-to-noise ratio (SNR) finds applications in economics and finance across various domains such as economic forecasting, financial modeling, market analysis and risk assessment. A high SNR indicates a robust and dependable signal, simplifying the process of making well-informed decisions. On the other hand, a low SNR indicates a weak signal that could be obscured by noise, so decision-making procedures need to take this into serious consideration. This research focuses on the development of confidence intervals for functions derived from the SNR and explores their application in the fields of economics and finance.Design/methodology/approachThe construction of the confidence intervals involved the application of various methodologies. For the SNR, confidence intervals were formed using the generalized confidence interval (GCI), large sample and Bayesian approaches. The difference between SNRs was estimated through the GCI, large sample, method of variance estimates recovery (MOVER), parametric bootstrap and Bayesian approaches. Additionally, confidence intervals for the common SNR were constructed using the GCI, adjusted MOVER, computational and Bayesian approaches. The performance of these confidence intervals was assessed using coverage probability and average length, evaluated through Monte Carlo simulation.FindingsThe GCI approach demonstrated superior performance over other approaches in terms of both coverage probability and average length for the SNR and the difference between SNRs. Hence, employing the GCI approach is advised for constructing confidence intervals for these parameters. As for the common SNR, the Bayesian approach exhibited the shortest average length. Consequently, the Bayesian approach is recommended for constructing confidence intervals for the common SNR.Originality/valueThis research presents confidence intervals for functions of the SNR to assess SNR estimation in the fields of economics and finance.

The Simultaneous Confidence Interval for the Ratios of the Coefficients of Variation of Multiple Inverse Gaussian Distributions and Its Application to PM2.5 Data

Due to slash/burn agricultural activity and frequent forest fires, PM2.5 has become a significant air pollution problem in Thailand, especially in the north and north east regions. Since its dispersion differs both spatially and temporally, estimating PM2.5 concentrations discretely by area, for which the inverse Gaussian distribution is suitable, can provide valuable information. Herein, we provide derivations of the simultaneous confidence interval for the ratios of the coefficients of variation of multiple inverse Gaussian distributions using the generalized confidence interval, the Bayesian interval based on the Jeffreys’ rule prior, the fiducial interval, and the method of variance estimates recovery. The efficacies of these methods were compared by considering the coverage probability and average length obtained from simulation results of daily PM2.5 datasets. The findings indicate that in most instances, the fiducial method with the highest posterior density demonstrated a superior performance. However, in certain scenarios, the Bayesian approach using the Jeffreys’ rule prior for the highest posterior density yielded favorable results.

Estimation of the percentile of Birnbaum-Saunders distribution and its application to PM2.5 in Northern Thailand.

The Birnbaum-Saunders distribution plays a crucial role in statistical analysis, serving as a model for failure time distribution in engineering and the distribution of particulate matter 2.5 (PM2.5) in environmental sciences. When assessing the health risks linked to PM2.5, it is crucial to give significant weight to percentile values, particularly focusing on lower percentiles, as they offer a more precise depiction of exposure levels and potential health hazards for the population. Mean and variance metrics may not fully encapsulate the comprehensive spectrum of risks connected to PM2.5 exposure. Various approaches, including the generalized confidence interval (GCI) approach, the bootstrap approach, the Bayesian approach, and the highest posterior density (HPD) approach, were employed to establish confidence intervals for the percentile of the Birnbaum-Saunders distribution. To assess the performance of these intervals, Monte Carlo simulations were conducted, evaluating them based on coverage probability and average length. The results demonstrate that the GCI approach is a favorable choice for estimating percentile confidence intervals. In conclusion, this article presents the results of the simulation study and showcases the practical application of these findings in the field of environmental sciences.

Bayesian Simultaneous Credible Intervals for the Differences of Coefficients of Variation of Weibull Distributions with Application to Wind Speed Data in Thailand

The Weibull distribution serves as a versatile tool for modelling various types of data, including reliability analysis, failure rates, survival analysis, and extreme value analysis. Specifically, in weather forecasting, the Weibull distribution is frequently used to represent variables such as wind speed. To examine differences in wind speed dispersion across different areas, the coefficient of variation proves optimal. However, when comparing variations across multiple areas simultaneously, the focus shifts to simultaneous confidence intervals (SCIs). Therefore, this paper focuses on the problem of constructing the SCIs for the differences of coefficients of variation derived from Weibull distributions. The proposed methods in this study are Bayesian approaches that utilize gamma and uniform prior distributions, namely the generalized confidence interval (GCI) and the method of variance estimate recovery (MOVER) based on the Hendricks and Robey interval. To assess the performance of these methods, their coverage probability, expected length, and standard errors were evaluated through a simulation study. The simulation results show that the Bayesian credible interval based on the gamma prior performs satisfactorily in most cases. Finally, to illustrate the SCIs, we present an example concerning wind speed dispersion data in Southern Thailand.

Simultaneous Confidence Intervals for Ratios of the Percentiles of the Delta-Lognormal Distribution

Statistical percentiles are extremely important and valuable measurement tools in various contexts. They offer significant advantages when it comes to estimating a wide range of values across different fields, making them essential in statistical analysis and research. This importance motivates researchers to study the construction of simultaneous confidence intervals for ratios of percentiles of the delta-lognormal distribution. This research introduces four distinct methods for constructing such confidence intervals: the fiducial generalized confidence interval, the Bayesian-based highest posterior density credible interval using Jeffreys prior, Jeffreys rule prior, and Uniform prior. All methods are subjected to evaluation and comparison through coverage probabilities and expected lengths in Monte Carlo simulations. The simulation study shows that, overall, the Bayesian-based highest posterior density credible interval works better and more accurately than the fiducial generalized confidence interval. To further validate the findings of the simulation study, these methods are applied to real-world daily rainfall data collected from a river basin in Thailand.

Statistical inference for a competing failure model based on the Wiener process and Weibull distribution.

Competing failure models with degradation phenomena and sudden failures are becoming more and more common and important in practice. In this study, the generalized pivotal quantity method was proposed to investigate the modeling of competing failure problems involving both degradation and sudden failures. In the competing failure model, the degradation failure was modeled through a Wiener process and the sudden failure was described as a Weibull distribution. For point estimation, the maximum likelihood estimations of parameters $ \mu $ and $ \sigma^2 $ were provided and the inverse estimation of parameters $ \eta $ and $ \beta $ were derived. The exact confidence intervals for parameters $ \mu $, $ \sigma^2 $, and $ \beta $ were obtained. Furthermore, the generalized confidence interval of parameter $ \eta $ was obtained through constructing the generalized pivotal quantity. Using the substitution principle, the generalized confidence intervals for the reliability function, the $ p $th percentile of lifetime, and the mean time to failure were also obtained. Simulation technique was carried out to evaluate the performance of the proposed generalized confidence intervals. The simulation results showed that the proposed generalized confidence interval was effective in terms of coverage percentage. Finally, an example was presented to illustrate the application of the proposed method.

Interval Estimation of the Stress-Strength Reliability in Lehmann Family of Distributions

This paper presents a unified approach for computing confidence limits for stress–strength reliability when strength and stress are independent random variables following a distribution in Lehmann family. The generalized confidence interval and the bootstrap confidence intervals are obtained. Simulation studies are conducted to assess the performance of the proposed methods in terms of the estimated coverage probabilities and the length of the confidence intervals. An example is also provided for illustration.

Simultaneous Confidence Intervals for All Pairwise Differences between Means of Weibull Distributions

The Weibull distribution is a continuous probability distribution that finds wide application in various fields for analyzing real-world data. Specifically, wind speed data often adhere to the Weibull distribution. In our study, our aim is to compare the mean wind speed datasets from different areas in Thailand. To achieve this, we proposed simultaneous confidence intervals for all pairwise differences between the means of Weibull distributions. The generalized confidence interval (GCI), method of variance estimates recovery (MOVER), and a Bayesian approach, utilizing both gamma and uniform prior distributions, are proposed to construct simultaneous confidence intervals. Through simulations, we find that the Bayesian highest posterior density (HPD) interval using a gamma prior distribution demonstrates satisfactory performance, while the GCI proves to be a viable alternative. Finally, we applied these proposed approaches to real wind speed data in northeastern and southern Thailand to illustrate their effectiveness and practicality.

Parametric Bootstrap Confidence Interval Estimation for the Percentile and Difference between the Percentiles of Delta-Lognormal Distributions with Application to Rainfall Data in Thailand

In Thailand, flooding often occurs during the summer monsoon when many tropical storms affect the country. The motivation of this study was to plan for and mitigate the damage caused by flooding in the future. The confidence interval (CI) for the percentile of a precipitation dataset can be used to estimate the intensity of rainfall in a particular area whereas the CI for the difference between the percentiles of two datasets can be used to compare the rainfall intensities in two areas. To this end, the performances of several approaches to estimate the CI for the percentile and difference between the percentiles of delta-lognormal distributions were constructed and compared. These estimates were constructed based on the Bayesian (BS) and parametric bootstrap (PB) approaches, as well as two fiducial generalized confidence interval (FGCI) approaches. The performances of the methods were evaluated using Monte Carlo simulation, the results of which indicate that the PB approach for both CIs performed the best in all scenarios tested. Its suitability was confirmed via two illustrative examples using daily rainfall datasets for Chiang Mai and Lampang provinces in Thailand.

Accelerated degradation data analysis based on inverse Gaussian process with unit heterogeneity

The unit heterogeneity of products and the nonlinear parameter-stress relationship often exist in practice. Therefore, considering the unit heterogeneity, the nonlinear accelerated model and inverse Gaussian process are developed to depict the accelerated degradation data. On the other hand, this more realistic model leads a challenge to derive the model parameter interval estimation. Thereby, a novel two-step interval estimation method is proposed for the proposed accelerated degradation model. First, generalized confidence intervals of the parameters characterizing random effect are derived from the Cornish–Fisher expansion, and their cumulative distribution functions are obtained. Then, using the generalized pivotal quantity procedure, generalized confidence intervals of the accelerated model parameters are derived. In addition, generalized confidence intervals of predictive reliability indexes are derived to guide the practical application. Finally, simulation studies and two real examples on Spiral springs and integrated circuit devices are presented to demonstrate the implementation of the proposed method.

Confidence intervals for ratio of means of delta-lognormal distributions based on left-censored data with application to rainfall data in Thailand.

Thailand is a country that is prone to both floods and droughts, and these natural disasters have significant impacts on the country's people, economy, and environment. Estimating rainfall is an important part of flood and drought prevention. Rainfall data typically contains both zero and positive observations, and the distribution of rainfall often follows the delta-lognormal distribution. However, it is important to note that rainfall data can be censored, meaning that some values may be missing or truncated. The interval estimator for the ratio of means will be useful when comparing the means of two samples. The purpose of this article was to compare the performance of several approaches for statistically analyzing left-censored data. The performance of the confidence intervals was evaluated using the coverage probability and average length, which were assessed through Monte Carlo simulation. The approaches examined included several variations of the generalized confidence interval, the Bayesian, the parametric bootstrap, and the method of variance estimates recovery approaches. For (ξ1, ξ2) = (0.10,0.10), simulations showed that the Bayesian approach would be a suitable choice for constructing the credible interval for the ratio of means of delta-lognormal distributions based on left-censored data. For (ξ1, ξ2) = (0.10,0.25), the parametric bootstrap approach was a strong alternative for constructing the confidence interval. However, the generalized confidence interval approach can be considered to construct the confidence when the sample sizes are increase. Practical applications demonstrating the use of these techniques on rainfall data showed that the confidence interval based on the generalized confidence interval approach covered the ratio of population means and had the smallest length. The proposed approaches' effectiveness was illustrated using daily rainfall datasets from the provinces of Chiang Rai and Chiang Mai in Thailand.

Inference for the Johnson SB distribution

This article develops inferential procedures for the Johnson SB distribution and the associated stress-strength model. The maximum likelihood estimators, uniformly minimum variance unbiased estimators and generalized confidence intervals for the parameters of the Johnson SB distribution are presented. The objective Bayesian inference methods based on the Jeffreys and reference priors are derived. Moreover, the generalized confidence lower limits and Bayesian credible lower limits for the reliability of the stress-strength model are provided. The performance of the proposed methods is evaluated by Monte Carlo simulation. Finally, an example is used to illustrate the proposed procedures.

Bayesian Estimation for the Difference between Coefficients of Quartile Variation of Delta-Lognormal Distributions: An Application to Rainfall in Thailand

The coefficient of quartile variation is a valuable measure used to assess data dispersion when it deviates from a normal distribution or displays skewness. In this study, we focus specifically on the delta-lognormal distribution. The lognormal distribution is characterized by its asymmetrical nature and comprises exclusively positive values. However, when these values undergo a logarithmic transformation, they conform to a symmetrical (or normal) distribution. Consequently, this research aims to establish confidence intervals for the difference between coefficients of quartile variation within lognormal distributions incorporating zero values. We employ the Bayesian, generalized confidence interval, and fiducial generalized confidence interval methods to construct these intervals, involving data simulation using RStudio software. We evaluate the performance of these methods based on coverage probabilities and average lengths. Our findings indicate that the Bayesian method, employing Jeffreys’ prior, performs well in low variability, while the generalized confidence interval method is more suitable for higher variability. Therefore, we recommend using both approaches to construct confidence intervals for the difference between the coefficients of the quartile variation in lognormal distributions that include zero values. Furthermore, we apply these methods to rainfall data in Thailand to illustrate their alignment with actual and simulated data.

Estimating average wind speed in Thailand using confidence intervals for common mean of several Weibull distributions.

The Weibull distribution has been used to analyze data from many fields, including engineering, survival and lifetime analysis, and weather forecasting, particularly wind speed data. It is useful to measure the central tendency of wind speed data in specific locations using statistical parameters for instance the mean to accurately forecast the severity of future catastrophic events. In particular, the common mean of several independent wind speed samples collected from different locations is a useful statistic. To explore wind speed data from several areas in Surat Thani province, a large province in southern Thailand, we constructed estimates of the confidence interval for the common mean of several Weibull distributions using the Bayesian equitailed confidence interval and the highest posterior density interval using the gamma prior. Their performances are compared with those of the generalized confidence interval and the adjusted method of variance estimates recovery based on their coverage probabilities and expected lengths. The results demonstrate that when the common mean is small and the sample size is large, the Bayesian highest posterior density interval performed the best since its coverage probabilities were higher than the nominal confidence level and it provided the shortest expected lengths. Moreover, the generalized confidence interval performed well in some scenarios whereas adjusted method of variance estimates recovery did not. The approaches were used to estimate the common mean of real wind speed datasets from several areas in Surat Thani province, Thailand, fitted to Weibull distributions. These results support the simulation results in that the Bayesian methods performed the best. Hence, the Bayesian highest posterior density interval is the most appropriate method for establishing the confidence interval for the common mean of several Weibull distributions.

Confidence Intervals for Mean and Difference between Means of Delta-Lognormal Distributions Based on Left-Censored Data

A delta-lognormal distribution consists of zero and positive values. The positive values follow a lognormal distribution, which is an asymmetric distribution. It is well known that the logarithm of these values follows a normal distribution, which is a symmetric distribution. The delta-lognormal distribution is used in medical and environmental sciences. This study considers the challenges of constructing confidence intervals for the mean and difference between means of delta-lognormal distributions containing left-censored data and applies them to compare two daily rainfall average areas in Thailand. Three different approaches for constructing confidence intervals for the mean of the delta-lognormal distribution containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, and the parametric bootstrap approach, are developed. Moreover, four different approaches for constructing confidence intervals for the difference between means of delta-lognormal distributions containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, the parametric bootstrap approach, and the method of variance estimates recovery approach, are considered. The performance of the proposed confidence intervals is evaluated by Monte Carlo simulation. The simulation studies indicate that the Bayesian approach can be considered as an alternative to construct a credible interval for the mean of the delta-lognormal distribution containing left-censored data. Additionally, the generalized confidence interval and Bayesian approaches can be recommended as alternatives to estimate the confidence interval for the difference between means of delta-lognormal distributions containing left-censored data. All approaches are illustrated using the daily rainfall data from Chiang Mai and Lampang provinces in Thailand.

Applications of process capability indices for suppliers selection problems using generalized confidence interval

In process capability indices, the generalized confidence interval (GCI) method has been used many times in several research articles. In this article, we have considered the supplier’s selection problems by using the GCI of difference between two process capability indices . We have considered two classical methods of estimation, viz. maximum likelihood estimation (MLE), and maximum product spacing estimation (MPSE) to estimate . By using Monte Carlo simulation, we have obtained the biases and corresponding mean squared errors (MSEs) of . Here, we also find the lower confidence limit (L), upper confidence limit (U), and their corresponding average width (AW) for both classical methods MLE and MPSE. Three real data sets have been reanalyzed to illustrate the methodology of the supplier’s selection problem by utilizing the generalized confidence interval method.

Uses of a new asymmetric loss-based process capability index in the electronic industries

This article suggests a novel process capability index (PCI) termed as , which is based on an asymmetric loss function (linear exponential) for a normal process and offers a specific method of incorporating the loss in capability analysis. Next, we estimate the suggested PCI using the moment estimation approach when the process follows a normal distribution, and we compare the effectiveness of the investigated estimation methods in terms of their mean squared errors through simulation analysis. Additionally, the confidence intervals for the index are constructed using the generalized confidence interval (GCI) and parametric bootstrap confidence interval (BCI) approach. Using Monte Carlo simulation, the performance of the GCI and BCI is compared in terms of average width, associated coverage probabilities, and relative coverage. Finally, three real data sets from the electronic industries are re-analyzed to show the usefulness of the suggested index, MOM estimation, GCI and BCI.

Confidence Interval Estimation for the Ratio of the Percentiles of Two Delta-Lognormal Distributions with Application to Rainfall Data

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently observed in the fields of environmental studies, biology, and medicine. The number of zero observations follows a binomial distribution. However, problems can arise in the analysis of data containing zero observations along with log-normally distributed data, for which the delta-lognormal distribution is often referred to for using the analysis of the data. In statistics, the percentile provides the relative standing of a numerical data point when compared to all of the others in a distribution with reference to the observations at or below it. In this study, estimates for the confidence interval for the ratio of the percentiles of two delta-lognormal distributions are constructed using fiducial generalized confidence interval approaches based on the fiducial quantity and the optimal generalized fiducial quantity, the Bayesian approach, and the parametric bootstrap method. As assessed by Monte Carlo simulations using the RStudio programming in terms of the coverage probability and the average length, the Bayesian approach performed quite well by providing adequate coverage probabilities along with the shortest average lengths in all of the scenarios tested. Daily rainfall data contain both zero and positive values. The daily rainfall data can usually be fitted to the delta-lognormal distribution. Their application to rainfall data is also provided to illustrate their efficacies with real data. The efficacy of the approach is used to compare two rainfall dispersion populations.

Confidence intervals for the difference between coefficients of variation of zero-inflated gamma distributions

<abstract><p>The problem of constructing confidence intervals (CIs) for the difference between coefficients of variation of two zero-inflated gamma distributions was considered. As gamma distribution does not have closed form maximum likelihood estimators, the parameters of gamma distribution have to be estimated numerically. To this end, we proposed here four different generalized confidence intervals (GCIs) based on fiducial inference, Box-Cox transformation, parametric bootstrap and the method of variance of estimates recovery (MOVER). Performances of the four GCIs were evaluated and compared via extensive simulation. The simulation results showed that all four methods returned satisfactory results according to coverage probabilities, even for the setting of small sample sizes.</p></abstract>