Generalized Confidence Interval Research Articles

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>

Reliability inference and remaining useful life prediction for the doubly accelerated degradation model based on Wiener process

<abstract><p>Degradation data are an important source of products' reliability information. Though stochastic degradation models have been widely used for fitting degradation data, there is a lack of efficient and accurate methods to get their confidence intervals, especially in small sample cases. In this paper, based on the Wiener process, a doubly accelerated degradation test model is proposed, in which both the drift and diffusion parameters are affected by the stress level. The point estimates of model parameters are derived by constructing a regression model. Furthermore, based on the point estimates of model parameters, the interval estimation procedures are developed for the proposed model by constructing generalized pivotal quantities. First, the generalized confidence intervals of model parameters are developed. Second, based on the generalized pivotal quantities of model parameters, using the substitution method the generalized confidence intervals for some interesting quantities, such as the degradation rate $ \mu_0 $, the diffusion parameter $ \sigma_0^2 $, the reliability function $ R(t_0) $ and the mean lifetime $ E(T) $, are obtained. In addition, the generalized prediction intervals for degradation amount $ X_0(t) $ and remaining useful life at the normal use stress level are also developed. Extensive simulations are conducted to investigate the performances of the proposed generalized confidence intervals in terms of coverage percentage and average interval length. Finally, a real data set is given to illustrate the proposed model.</p></abstract>

Estimation for inverse Weibull distribution under progressive type-Ⅱ censoring scheme

<abstract><p>This paper considers the statistical inferences of inverse Weibull distribution under progressive type-Ⅱ censored sample, which is a common distribution in reliability analysis. Two commonly used parameter estimation methods, maximum likelihood estimation and Bayesian estimation, are used in this paper, along with the inverse moment estimation. First, we derive the maximum likelihood estimators of parameters and propose Newtown-Raphson iteration method to solve these estimators. Assuming that shape and rate parameters are independent and follow gamma priors, we further obtain the Bayesian estimators by Lindley approximation. We also derive the inverse moment estimators and construct the generalized confidence intervals using the generalized pivotal quantity. To compare the estimation effects of these methods, we implement Monte Carlo simulation with the help of MATLAB. The simulation results show that the Bayesian estimation method outperforms the other two methods in terms of mean squared error. Finally, we verify the feasibility of these methods by analyzing a set of real data. The results indicate that the Bayesian estimation method provides more accurate estimates than the other two methods.</p></abstract>

Confidence Interval Estimation for the Common Mean of Several Zero-Inflated Gamma Distributions

In this study, we propose estimates for the confidence interval for the common mean of several zero-inflated gamma (ZIG) distributions based on the fiducial generalized confidence interval (GCI) and Bayesian and highest posterior density (HPD) methods based on the Jeffreys rule or uniform prior. Their performances in terms of their coverage probabilities and expected lengths are compared via a Monte Carlo simulation study. For almost all of the scenarios considered, the simulation results show that the fiducial GCI performed better than the Bayesian and HPD methods. Daily rainfall data from Chiang Mai Province, Thailand that contains several zero entries and follows a ZIG distribution is used to test the efficacies of the methods in real-world situations.

Interval estimation of the two-parameter exponential constant stress accelerated life test model under Type-II censoring

ABSTRACT In this paper, interval estimation of the two-parameter exponential distribution based on constant-stress accelerated life test data under Type-II censoring is studied. The life-stress models of the scale and location parameters are assumed to be log-linear. The maximum likelihood estimators of the model parameters are obtained. The exact and approximate confidence intervals for the parameters in the life-stress model of the scale parameter are developed. The generalized confidence intervals for the parameters in the life-stress model of the location parameter, and mean and reliability function at the designed stress level are derived. The performance of the proposed generalized confidence intervals is assessed by the Monte Carlo simulation. Finally, a real example is used to illustrate our methods.

Estimation of the Confidence Interval for the Ratio of the Coefficients of Variation of Two Weibull Distributions and Its Application to Wind Speed Data

The Weibull distribution, one of the most significant distributions with applications in numerous fields, is associated with numerous distributions such as generalized gamma distribution, exponential distribution, and Rayleigh distribution, which are asymmetric. Nevertheless, it shares a close relationship with a normal distribution where a process of transformation allows them to become symmetric. The Weibull distribution is commonly used to study the failure of components and phenomena. It has been applied to a variety of scenarios, including failure time, claims amount, unemployment duration, survival time, and especially wind speed data. A suitable area for installing a wind turbine requires a wind speed that is both sufficiently high and consistent, and so comparing the variation in wind speed in two areas is eminently desirable. In this paper, methods to estimate the confidence interval for the ratio of the coefficients of variation of two Weibull distributions are proposed and applied to compare the variation in wind speed in two areas. The methods are the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER), and Bayesian methods based on the gamma and uniform priors. The Bayesian methods comprise the equal-tailed confidence interval and the highest posterior density (HPD) interval. The effectiveness of the methods was evaluated in terms of their coverage probabilities and expected lengths and also empirically applied to wind speed datasets from two different areas in Thailand. The results indicate that the HPD interval based on the uniform prior outperformed the others in most of the scenarios tested and so it is suggested for estimating the confidence interval for the ratio of the coefficients of variation of two Weibull distributions.

Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions

In situations where several positive random variables cannot be described using symmetrical distributions, a positively asymmetric distribution which has garnered much attention for studying them is the Birnbaum-Saunders (BS) distribution. This distribution was originally proposed to study fatigue over time in materials and has become widely employed for reliability and fatigue studies. In statistics, the coefficient of variation (CV) is employed to measure relative variation. Furthermore, comparing the CVs of several samples from BS distributions is an important approach to assess the variation among them. Herein, we propose estimation methods for the simultaneous confidence intervals (SCIs) for all pairwise differences between the CVs of multiple BS distributions based on the percentile bootstrap, the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER) based on the asymptotic confidence interval (ACI) and GCI, Bayesian credible interval, and the highest posterior density (HPD) interval. The coverage probabilities and average lengths of the proposed methods were examined via a simulation study to determine their performance. The results demonstrate that GCI and the MOVER based on the GCI method provided satisfactory performances in almost every case studied. Particulate matter ≤ 2.5 μm (PM2.5) concentration datasets from three areas in northern Thailand were used to illustrate the effectiveness of the proposed methods.

Estimation of common percentile of rainfall datasets in Thailand using delta-lognormal distributions.

Weighted percentiles in many areas can be used to investigate the overall trend in a particular context. In this article, the confidence intervals for the common percentile are constructed to estimate rainfall in Thailand. The confidence interval for the common percentile help to indicate intensity of rainfall. Herein, four new approaches for estimating confidence intervals for the common percentile of several delta-lognormal distributions are presented: the fiducial generalized confidence interval, the adjusted method of variance estimates recovery, and two Bayesian approaches using fiducial quantity and approximate fiducial distribution. The Monte Carlo simulation was used to evaluate the coverage probabilities and average lengths via the R statistical program. The proposed confidence intervals are compared in terms of their coverage probabilities and average lengths, and the results of a comparative study based on these metrics indicate that one of the Bayesian confidence intervals is better than the others. The efficacies of the approaches are also illustrated by applying them to daily rainfall datasets from various regions in Thailand.

Objective Bayesian analysis of the ratio of bivariate lognormal means

In this study, we consider the objective Bayesian analysis for the ratio of means in the bivariate lognormal distribution, in which properly assigning priors for the ratio of means is challenging due to the presence of nuisance parameters. As a result, we create probability matching priors and reference priors for the ratio of means. Jeffreys’ prior and the two group reference prior do not satisfy a first-order matching criterion. We also investigate the posterior distribution property for the general prior class, which includes Jeffreys’ prior, the reference prior, and the matching prior. Through a simulation study, we check the frequentist coverage probabilities and compare them with the generalized confidence interval method. It demonstrated that the proposed probability matching priors match very well with the target coverage probabilities even when the sample sizes are small in a frequentist sense. Two real examples are also provided.

Confidence Intervals for Common Coefficient of Variation of Several Birnbaum–Saunders Distributions

The Birnbaum–Saunders (BS) distribution, also known as the fatigue life distribution, is right-skewed and used to model the failure times of industrial components. It has received much attention due to its attractive properties and its relationship to the normal distribution (which is symmetric). Furthermore, the coefficient of variation (CV) is commonly used to analyze variation within a dataset. In some situations, the independent samples are collected from different instruments or laboratories. Consequently, it is of importance to make inference for the common CV. To this end, confidence intervals based on the generalized confidence interval (GCI), method of variance estimates recovery (MOVER), large-sample (LS), Bayesian credible interval (BayCrI), and highest posterior density interval (HPDI) methods are proposed herein to estimate the common CV of several BS distributions. Their performances in terms of their coverage probabilities and average lengths were investigated by using Monte Carlo simulation. The simulation results indicate that the HPDI-based confidence interval outperformed the others in all of the investigated scenarios. Finally, the efficacies of the proposed confidence intervals are illustrated by applying them to real datasets of PM10 (particulate matter ≤ 10 μm) concentrations from three pollution monitoring stations in Chiang Mai, Thailand.

Generalized confidence intervals of quantile-based process capability indices for inverse Gaussian distribution

ABSTRACT Process capability index (PCI) is widely used to evaluate the process capability in various industries. In this article, generalized confidence intervals (GCIs) of two widely used percentile-based PCIs for the inverse Gaussian (IG) distribution are constructed. Meanwhile, the GCI of the difference between two IG processes’ PCIs is also considered. In addition, the performance of the proposed GCI is evaluated and compared with that of the confidence intervals based on the traditional bootstrap method. The simulation results indicate that the GCI outperforms the bootstrap counterparts in terms of the coverage probabilities, and the actual coverage probabilities of the GCI are sufficiently close to the target value. Finally, two real examples are presented to illustrate the implementation of the proposed method.

Assessment of Two Process Capabilities by Using Generalized Confidence Intervals and its Applications

Assessment of Two Process Capabilities by Using Generalized Confidence Intervals and its Applications

Inference for constant stress accelerated life test under the proportional reverse hazards lifetime distribution

AbstractIn this paper, we consider small sample statistical inference methods for the constant stress accelerated life test (CSALT) under the proportional reverse hazards family (PRHF). The model parameters are assumed to be the exponential functions of the stress. The generalized confidence intervals (GCIs) are derived for the model parameters and some commonly used reliability metrics such as the quantile and the reliability function of the lifetime at the designed stress level. The generalized prediction interval (GPI) is also proposed for the future failure time (FFT) and the remaining useful life (RUL) at the designed stress level. The Monte Carlo simulation study for the generalized Lindley distribution (GLD) and the inverse Weibull distribution are used to illustrate the performance of the proposed GCIs and GPI. The simulation results indicate that the performance of the proposed GCIs and GPI is better than the Wald confidence intervals (CIs) and bootstrap‐p CIs in terms of the coverage probability (CP). Finally, a real example is used to illustrate the developed procedures.