Generalized Confidence Interval Method Research Articles

Confidence Intervals for the Mean and Difference of Means of Birnbaum-Saunders Distributions with Application to Wind Speed Data

This paper proposes the confidence intervals for the mean and difference of means of Birnbaum-Saunders (BirSau) distributions based on the Bootstrap confidence interval (BCI), Percentile bootstrap confidence interval (PBCI), Generalized confidence interval (GCI), Bayesian credible interval (BayCrI) and the highest posterior density (HPD). The simulation study used R statistical software to evaluate the coverage probabilities and average lengths. The concerning results of the mean suggest that HPD is the recommended method for constructing confidence intervals in the BirSau distributions, except for small sample sizes where the GCI method proves more efficient. For the difference of means, PBCI emerges as the preferred way to construct confidence intervals, except in some cases where small sample sizes with the HPD method are more efficient. Moreover, the average lengths of these proposed confidence intervals decreased as both sample size and shape parameters increased. To illustrate the effectiveness of the suggested confidence intervals, we applied them to wind speed datasets collected in Ayutthaya and Ratchaburi provinces, Thailand.

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.

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.

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.

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 the weighted coefficients of variation of two-parameter exponential distributions

This paper proposes new confidence intervals for the weighted coefficients of variation (CV) of two-parameter exponential distributions based on the adjusted method of variance estimates recovery method (adjusted MOVER). This is then compared with the generalized confidence interval method (GCI) and the large sample method. The performance of these confidence intervals in terms of coverage probabilities and average lengths were evaluated via a Monte Carlo simulation. Simulation studies showed that the GCI should be considered as an alternative to the confidence interval estimation for the weighted CV of two-parameter exponential distributions. However, the adjusted MOVER confidence interval (CIAM2) can be used to estimate the weighted CV when the coefficient of variation is a positive value. The proposed confidence intervals are illustrated using a real example.

Performance evaluation of processes with asymmetric tolerances in the presence of gauge measurement errors

ABSTRACTIn this study, the generalized confidence interval method is applied to evaluate the performances of processes with asymmetric tolerances taking into account the gauge measurement errors (GME). To examine the performance of the proposed method, a series of simulations is conducted. Moreover, a sensitivity study is carried out to analyze the effects of ignoring GME. The proposed method performs very well and can be recommended for assessing the performances of processes with asymmetric tolerances in the presence of GME.

Reliability evaluation for Weibull distribution under multiply type-І censoring

The multiply type-І censoring represented that all units in life test were terminated at different times. For estimations of Weibull parameters, it was easy to compute the maximum likelihood estimation (MLE) and least-squares estimation (LSE) while it was hard to build confidence intervals (CI). The concept of generalized confidence interval (GCI) was introduced to build CIs of parameters under multiply type-I censoring. Further, GCI based on LSE and GCI based on MLE were proposed. It is mathematically proved that the former is exact and the latter is approximate. Besides, a Monte Carlo simulation study and an illustrative example also turn out that the GCI method based on LSE yields rather satisfactory results by comparison with the ones based on MLE. It should be clear that the GCI method is a sensible choice to evaluate reliability under multiply type-I censoring.

Assessment of Cpm in the presence of measurement errors

With the advent of Taguchi's loss function, the concept of target and sticking to the target to achieve better process performance has become widely accepted. In practice, while estimating process performance, the gauge measurement error is not taken into consideration. In the real world scenario this hardly happens since measurement errors cannot be avoided in most of the manufacturing processes. Ignoring this measurement error while estimating the process capability may often lead to unreliable/wrong decision about the capability of the process under study. Therefore, in this work we apply the method of Generalized Confidence Interval (GCI) to measure the process capability index ܥ ௣௠ in presence of measurement errors. In this study, an exhaustive set of simulation has been conducted to assess the performance of the GCI method in terms of expected value of generalized lower confidence limit (ܮ ௣௠) and Coverage Probability (CP). The efficacy of dealing with the measurement error has been found satisfactory in this model. Finally it can be concluded that GCI method seems to be quite satisfactory for measuring process capability when the measurement errors are present; as well as when measurement error is negligible.

Comparison of confidence interval methods for an intra-class correlation coefficient (ICC).

BackgroundThe intraclass correlation coefficient (ICC) is widely used in biomedical research to assess the reproducibility of measurements between raters, labs, technicians, or devices. For example, in an inter-rater reliability study, a high ICC value means that noise variability (between-raters and within-raters) is small relative to variability from patient to patient. A confidence interval or Bayesian credible interval for the ICC is a commonly reported summary. Such intervals can be constructed employing either frequentist or Bayesian methodologies.MethodsThis study examines the performance of three different methods for constructing an interval in a two-way, crossed, random effects model without interaction: the Generalized Confidence Interval method (GCI), the Modified Large Sample method (MLS), and a Bayesian method based on a noninformative prior distribution (NIB). Guidance is provided on interval construction method selection based on study design, sample size, and normality of the data. We compare the coverage probabilities and widths of the different interval methods.ResultsWe show that, for the two-way, crossed, random effects model without interaction, care is needed in interval method selection because the interval estimates do not always have properties that the user expects. While different methods generally perform well when there are a large number of levels of each factor, large differences between the methods emerge when the number of one or more factors is limited. In addition, all methods are shown to lack robustness to certain hard-to-detect violations of normality when the sample size is limited.ConclusionsDecision rules and software programs for interval construction are provided for practical implementation in the two-way, crossed, random effects model without interaction. All interval methods perform similarly when the data are normal and there are sufficient numbers of levels of each factor. The MLS and GCI methods outperform the NIB when one of the factors has a limited number of levels and the data are normally distributed or nearly normally distributed. None of the methods work well if the number of levels of a factor are limited and data are markedly non-normal. The software programs are implemented in the popular R language.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2288-14-121) contains supplementary material, which is available to authorized users.

On Confidence Intervals for Process Capability Indices in a One-Way Random Model

In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C pk for the one-way random effect model. Also, we derived Bissell's approximation formula for the lower confidence limit using Satterthwaite's method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets.

Confidence Intervals in a Gauge R&R Study with Fixed Operators

The objective of this paper is to develop a method for constructing confidence intervals in a two-factor gauge repeatability and reproducibility (R&R) study with fixed operators. A previously proposed method is compared with a generalized confidence interval method using computer simulation. Simulation results suggest that, although the two methods are comparable in most cases, there are situations where the generalized confidence interval is a better choice.