Generalized Confidence Interval Research Articles

Confidence intervals of difference and ratio between two means of delta-Birnbaum-Saunders distributions

During the dry season, the northern region of Thailand consistently grapples with compromised air quality, notably an excessive level of particulate matter with a diameter of less than 2.5 microns (PM2.5), which poses implications for public health. Wind speed emerges as the primary factor that diminishes the concentration of PM2.5, making the evaluation of wind speed changes crucial in understanding and potentially mitigating the impact of air pollution on public health in the region. The wind speed data contain zero and positive values. This can be estimated through the confidence intervals of the difference and ratio between two means of delta-Birnbaum-Saunders distributions using various methods, including the generalized confidence interval (GCI), bootstrap confidence interval (BCI), and generalized fiducial confidence interval (GFCI). These methods rely on variance stabilizing transformations (VST), the Wilson and Hannig methods. According to the simulation, for small sample sizes, the GCI based on the VST and Hannig was the most effective method in terms of coverage probabilities and average lengths. For medium sample sizes, it was found that the performance of GFCI based on the VST method was superior to other methods. Additionally, when confronted with a large sample size, the BCI based on the VST method becomes relevant and will be introduced. To demonstrate the efficacy of our proposed methods, we applied them to wind speed data gathered from Chiang Mai Airport station in Thailand from January to February in both 2022 and 2023.

Estimation methods for the variance of Birnbaum-Saunders distribution containing zero values with application to wind speed data in Thailand.

Thailand is currently grappling with a severe problem of air pollution, especially from small particulate matter (PM), which poses considerable threats to public health. The speed of the wind is pivotal in spreading these harmful particles across the atmosphere. Given the inherently unpredictable wind speed behavior, our focus lies in establishing the confidence interval (CI) for the variance of wind speed data. To achieve this, we will employ the delta-Birnbaum-Saunders (delta-BirSau) distribution. This statistical model allows for analyzing wind speed data and offers valuable insights into its variability and potential implications for air quality. The intervals are derived from ten different methods: generalized confidence interval (GCI), bootstrap confidence interval (BCI), generalized fiducial confidence interval (GFCI), and normal approximation (NA). Specifically, we apply GCI, BCI, and GFCI while considering the estimation of the proportion of zeros using the variance stabilized transformation (VST), Wilson, and Hannig methods. To evaluate the performance of these methods, we conduct a simulation study using Monte Carlo simulations in the R statistical software. The study assesses the coverage probabilities and average widths of the proposed confidence intervals. The simulation results reveal that GFCI based on the Wilson method is optimal for small sample sizes, GFCI based on the Hannig method excels for medium sample sizes, and GFCI based on the VST method stands out for large sample sizes. To further validate the practical application of these methods, we employ daily wind speed data from an industrial area in Prachin Buri and Rayong provinces, Thailand.

The impact of spinal versus general anesthesia on the variability of surgical times: a systematic review and meta-analysis.

With spinal anesthesia, when cases are taking longer than usual, there may be behavioural tendencies for surgical teams to work more quickly. We conducted a systematic review with meta-analysis to examine standard deviations of surgical times for single-dose spinal anesthetics versus general anesthesia. We compared ratios of mean surgical times as a secondary endpoint. We included randomized trials of humans where general or spinal anesthesia was used for one category of surgical procedure (e.g., hip arthroplasty) and the article reported the means and standard deviations of operative durations. We used statistical methods suitable for surgical times following log-normal distributions. We used generalized confidence intervals to calculate point estimates of ratios and standard errors for each study, followed by pooling among studies using DerSimonian and Laird random-effects meta-analysis with Knapp-Hartung adjustment. Among the 77 included studies, 96% were of high quality for our endpoint (i.e., had a low risk of bias), as no (0%) study focused on comparing variability of surgical times and none had surgical time as the primary endpoint. Spinal anesthesia was associated with 6.6% smaller standard deviations than general anesthesia (95% confidence interval, 15.8% smaller to 1.9% larger, P = 0.13). By meta-regression, there was no significant association of the ratios of standard deviations with study quality (P = 0.39), year of publication (P = 0.76), or categories of procedures (all five P ≥ 0.28). Spinal anesthesia was associated with 1.1% smaller means than general anesthesia (95% confidence interval, 3.7% smaller to 1.5% larger, P = 0.42). There were no significant associations between the ratios of means and study quality (P = 0.47), year of publication (P = 0.95), or categories of procedures (all five, P ≥ 0.63). The results of this systematic review and meta-analysis show with high confidence that the effect of choosing spinal anesthesia on variability in surgical time, if present, is sufficiently small to have no substantive direct economic effect. The same conclusion applies to mean surgical time. Therefore, although anesthetic choice has a clinical (biological) impact and affects anesthesia times, the direct effects on surgical times and workflow are minimal at most. Anesthetic choice does not influence operating theatre productivity via changes to surgical times. The impact of spinal anesthetic effects is limited to nonoperative times (e.g., reducing anesthesia-controlled times by using a block room before the patient enters the operating room). PROSPERO ( CRD42023461952 ); first submitted 8 September 2023.

Likelihood and pivotal-based reliability inference for inverse exponentiated Rayleigh distribution under block progressive censoring

ABSTRACT In this paper, focus is on the estimation of survival characteristics, specifically the reliability function, hazard rate function, median time to failure, and differences in different test facilities, using block progressive censored data. Estimations are carried out through both maximum likelihood and pivotal methods, assuming the lifetime distribution of test units follows an inverse exponentiated Rayleigh distribution. The paper derives maximum likelihood estimators for unknown parameters, exploring their existence and uniqueness properties. Approximate confidence intervals for survival characteristics are constructed using the delta method and likelihood theory. Moreover, point and generalized confidence interval estimators are developed through a pivotal quantity-based method. A simulation study is conducted to compare the performance of the proposed approaches, revealing that the pivotal quantity-based approach yields superior estimation results. Finally, the proposed estimates are applied to analyze two real datasets, illustrating their practical applicability.

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.

Generalized Confidence Intervals for Ratios of Standard Deviations Based on Log-Normal Distribution when Times Follow Weibull Distributions.

Modern anesthetic drugs ensure the efficacy of general anesthesia. Goals include reducing variability in surgical, tracheal extubation, post-anesthesia care unit, or intraoperative response recovery times. Generalized confidence intervals based on the log-normal distribution compare variability between groups, specifically ratios of standard deviations. The alternative statistical approaches, performing robust variance comparison tests, give P-values, not point estimates nor confidence intervals for the ratios of the standard deviations. We performed Monte-Carlo simulations to learn what happens to confidence intervals for ratios of standard deviations of anesthesia-associated times when analyses are based on the log-normal, but the true distributions are Weibull. We used simulation conditions comparable to meta-analyses of most randomized trials in anesthesia, and coefficients ofvariation . The estimates of the ratios of standard deviations were positively biased, but slightly, the ratios being 0.11% to0.33% greater than nominal. In contrast, the 95% confidence intervals were very wide (i.e., > 95% of P ≥ 0.05). Although substantive inferentially, the differences inthe confidence limits were small from a clinical or managerial perspective, with amaximum absolute difference inratios of 0.016. Thus, P < 0.05 isreliable, but investigators should plan for TypeII errors atgreater than nominal rates.

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.

Generalized Confidence Interval for the Difference Between Percentiles of Birnbaum–Saunders Distributions and Its Application to PM2.5 in Thailand

The Birnbaum–Saunders distribution is of particular interest for statistical inference. This distribution represents the failure time distribution in engineering. In addition, the Birnbaum–Saunders distribution is commonly used in different areas of science and engineering. Percentiles are a frequently employed statistical concept. Percentiles help ascertain the position of an observation concerning the percentage of data points below it. These percentiles serve as indicators of both the central tendency and the dispersion of data. While comparing two data distributions, the mean is typically the most dependable parameter for describing the population. However, in situations where the distribution exhibits significant skewness, percentiles may sometimes offer a more reliable representation. Herein, the confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions were constructed by the generalized confidence interval (GCI) approach, the bootstrap approach, the Bayesian approach, and the highest posterior density (HPD) approach. A Monte Carlo simulation was conducted to evaluate the performance of the confidence intervals. The performance was considered via coverage probability and average width. The findings suggest that utilizing the GCI approach is advisable for estimating confidence intervals for the disparity between two percentiles. Ultimately, the outcomes of the simulation investigation, coupled with an application in the field of environmental sciences, were outlined.

Accurate inference for the Youden index and its associated cutoff point based on the gamma and inverse Gaussian distributed assumption

&lt;p&gt;The Youden index is often used to measure the effectiveness of biomarkers and aids to find the optimal cutoff point. Since pooled specimens have been shown to be an effective cost-cutting technique, we proposed the exact inferential procedures for the Youden index and its associated cutoff point based on the pooled specimens under the gamma or the inverse Gaussian assumption. The generalized confidence intervals (GCIs) were proposed for the Youden index and its associated cutoff point. Monte Carlo simulations were used to assess the performance of the proposed GCIs. The simulation results show that the proposed GCIs outperformed existing methods such as the bootstrap-$ p $ CIs in terms of the coverage probability. Finally, the proposed procedures were illustrated by an example.&lt;/p&gt;

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.