Highest Posterior Density Credible Intervals Research Articles

Objective Bayesian estimation for multistate stress‐strength model's reliability with various kernel functions

AbstractThe paper discusses the objective Bayesian estimation of the reliability of a multistate stress‐strength model (MSSM) based on different kernel functions. For the MSSM, we first derive the reliability function and Fisher information matrix. The Jeffreys prior, reference prior, and probability matching prior for the reliability function of the MSSM are constructed based on the objective Bayesian paradigm. Subsequently, we demonstrated that these priors are improper density, then evaluated the effects of these priors on Bayes estimates for MSSM's reliability based on a complete sample. The Bayesian estimates are calculated using random walk Metropolis–Hastings techniques. We employ Monte Carlo simulation to examine the effectiveness of Bayes estimates for MSSM's reliability in terms of average bias and mean squared error, meanwhile the highest posterior density credible intervals are investigated in terms of average length and coverage probability. Finally, two real datasets were examined, demonstrating the viability of the objective Bayes technique for small sample data.

Estimation of stress-strength reliability in s-out-of-k system for new flexible exponential distribution under progressive type-II censoring

This article addresses stress-strength reliability estimation in s-out-of-k systems under progressive type-II censoring.  The system consists of components facing common stress, where stress and strength variable follows independent new flexible exponential (NFE) distributions, sharing a common parameter. The estimation procedure is carried out using MLE, Bayesian estimation and uniformly minimum variance unbiased estimation (UMVUE) techniques. MLE is derived for known and unknown common parameters. Bayesian estimates are obtained using Lindley’s approximation and Markov chain Monte Carlo techniques due to the complexity of obtaining the exact Bayesian estimates, when the common parameter is unknown. Moreover, UMVUE is derived when the common parameter is known. The 95% asymptotic confidence interval, highest posterior density credible intervals, and Bayesian credible intervals, are constructed. An extensive simulation study is carried out to compare the performance of various censoring schemes and estimation methods. Finally, two progressively censored real datasets are analyzed for illustration.

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.

Randomly Censored Kumaraswamy Distribution

In this paper, inferential procedures based on classical and Bayesian framework for the Kumaraswamy distribution under random censoring model are studied. We first propose estimators for the distribution parameters, reliability function, failure rate function, and Mean time to system failure based on the maximum likelihood estimation method. Then, we calculate asymptotic confidence intervals for the parameters based on the observed Fisher’s information matrix. Also, for the parameters and reliability characteristics, Bayesian estimates are derived using the importance sampling and Gibbs sampling procedures. Highest posterior density credible intervals for the parameters are constructed using Markov Chain Monte Carlo method. Expected time on test of experiment with random censoring is also calculated. A simulation study is conducted to compare the efficiency of the derived estimates. Finally, the analysis of a real data set is presented for the illustration purpose.

Inference of reliability model with burr type XII distribution under two sample balanced progressive censored samples

Abstract The balanced joint progressive censoring scheme gives guidelines for researchers in reducing the expense and period of the experiment with modified efficiency. In this paper, the stress-strength reliability model are considered and investigated, in which two independent samples are generated from Burr Type XII distributions under balanced joint progressive censoring scheme. In the classical analysis, the maximum likelihood estimate of the reliability parameter is derived, then the relevant confidence interval is provided based on the Fisher matrix. Under the Bayesian framework, the Lindley’s technique and hybrid Gibbs within Metropolis-Hastings algorithm are applied to acquire the estimates of the parameters, and the corresponding highest posterior density credible interval of the reliability model is also constructed. The reliability inference is realized by numerical simulation, and the validity of the inference method is analyzed based on the splashing data of silicone oil droplets impact (SSODI) under two surfaces wettability conditions.

Statistical inference of the Birnbaum-Saunders model using adaptive progressively hybrid censored data and its applications

<abstract><p>Lately, the Birnbaum-Saunders distribution has gained a lot of attention, mainly due to its different density shapes and the non-monotonicity property of its failure rates. This work considered some estimation issues for the Birnbaum-Saunders distribution using adaptive progressive Type-Ⅱ hybrid censoring. Point and interval estimations were performed employing both conventional and Bayesian methodologies. In addition to estimating the model parameters, we obtained point and interval estimates for the reliability and hazard rate functions. We looked at the method of maximum likelihood as a classical approach, and its asymptotic traits were employed to obtain approximate confidence ranges. From a Bayesian point of perspective, we considered the squared error loss function to obtain the point estimates of the various parameters. The Bayes and highest posterior density credible intervals were additionally determined. For the complex form of the posterior distribution, Bayes estimates and credible intervals were computed by sampling from the posterior distribution through the Markov chain Monte Carlo procedure. For assessing the performance of all of these estimators, a Monte Carlo simulation was employed. Several statistical standards were applied to check the effectiveness of various estimates for multiple levels of censoring with small, moderate, and large sample sizes. Finally, two scenarios for applications were given in order to highlight the usefulness of the supplied approaches.</p></abstract>

Statistical inference for the step-stress model with competing risks from the Kumaraswamy distribution under progressive type-II censoring

In reliability and life analysis, the accelerated life test is frequently used because it can reduce cost and obtain additional reliability and lifetime information. In addition, when a unit fails in a life test, there are often two or more risk factors related to the cause of the failure. In this article, we consider the inference from a step-stress-accelerated life test with competing risks using progressive type-II censored data. Based on the assumption that the parameters affected by stress follow a log-linear model with the stress level, the proportional hazard model with the Kumaraswamy distribution is established. The point estimation of the unknown parameters is derived using maximum likelihood and Bayesian methods. Accordingly, the logarithmic asymptotic confidence and the highest posterior density credible intervals are derived and constructed. Moreover, the algorithm for multi-parameter sampling and simulation technology based on this model is given. Simulation results show that the proposed methods have good performance. In light of the estimates of the parameters, the estimated reliability function under normal conditions can be expressed, and its images under different methods are drawn. Last, a real dataset and a set of simulated data are presented for illustrative purposes.

Bayesian Analysis of Unit Log-Logistic Distribution Using Non-Informative Priors

The unit log-logistic distribution is a suitable choice for modeling data enclosed within the unit interval. In this paper, estimating the parameters of the unit-log-logistic distribution is performed through a Bayesian approach with non-informative priors. Specifically, we use Jeffreys, reference, and matching priors, with the latter depending on the interest parameter. We derive the corresponding posterior distributions and validate their propriety. The Bayes estimators are then computed using Markov Chain Monte Carlo techniques. To assess the finite sample performance of these Bayes estimators, we conduct Monte Carlo simulations, evaluating their mean squared errors and their coverage probabilities of the highest posterior density credible intervals. Finally, we use these priors to obtain estimations and credible sets for the parameters in an example of a real data set for illustrative purposes.

Parameter Estimation of Inverted Exponentiated Half-Logistic Distribution under Progressive Type-II Censored Data with Competing Risks

In this work, the statistical inference of inverted exponentiated half-logistic distribution is considered under the progressive type-II censored data with competing risks. The parameter estimations of the constant scale parameters but different shape parameters that changed due to various risk factors are investigated in this study. The maximum likelihood estimation is deduced to obtain the parameter estimators and asymptotic confidence intervals; meanwhile, the existence and uniqueness of the maximum likelihood estimators are discussed. Subsequently, bootstrap confidence intervals are calculated and presented. The Bayes estimates based on the square error loss function are derived using the Markov Chain Monte Carlo method. Then, the highest posterior density credible intervals are calculated and provided. Simulations are implemented to compare two kinds of estimates. For illustrative purposes, two experimental lifetime datasets from different fields are exhibited and analyzed. Finally, an optimal censoring scheme is suggested under two optimality criteria.

Bayesian and likelihood estimation of multicomponent stress–strength reliability from power Lindley distribution based on progressively censored samples

In this article, the problem of estimation of reliability of a ℓ-component system when both the stress and strength components are assumed to have a power Lindley distribution is discussed. The multicomponent stress–strength reliability parameter is obtained using both the Bayesian and the classical approaches when component-wise each unit follows a power Lindley distribution. To estimate the multicomponent stress–strength reliability parameter under the classical approach, the method of maximum likelihood and the asymptotic confidence interval estimation method are used as point and interval estimation methods, respectively. Under the Bayesian paradigm, the reliability parameter is estimated under the linear exponential loss function using the Lindley approximation, the Tierney–Kadane approximation and the Markov chain Monte Carlo (MCMC) techniques and subsequently highest posterior density credible intervals are obtained. To validate the efficacy of the proposed estimation strategies, a simulation study is carried out. Finally, two real-life data sets are re-analysed for illustrative purposes.

On competing risk model under step-stress stage life testing

In this article, we consider the inference of a dependent competing risk model when the data are collected from a simple step-stress stage life testing experiment. We assume that there are two causes of failure and they are dependent causes. The dependency structure is explained through Marshall–Olkin bivariate Weibull distribution. The distribution under two stress levels is connected through Khamis Higgins model. We obtain the maximum likelihood estimates and the Bayes estimates of the model parameters. We also provide the asymptotic confidence intervals, symmetric credible intervals and the highest posterior density credible intervals of the model parameters. An optimality study has been done under different optimality criteria. An extensive simulation study and a data analysis have been performed for illustrative purpose.

Bayesian Estimation and Prediction for Inverse Power Maxwell Distribution with Applications to Tax Revenue and Health Care Data

This article discusses the problem of estimating the unknown model parameters as well as prediction of future observations from inverse power Maxwell distribution. The maximum likelihood method is applied for estimating the model parameters using Newton-Rapson iterative procedures. The existence and uniqueness of maximum likelihood estimates are established using Cauchy-Schwartz inequality. Approximate confidence intervals are constructed using Fisher information matrix. Using independent gamma informative priors, the Bayes estimates of unknown model parameters are obtained under squared error and Linex loss functions. Two approximation techniques namely: Lindley’s approximation and Metropolis-Hastings within Gibbs sampler algorithm have been employed to derive the Bayes estimators and also to construct the associate highest posterior density credible intervals. Based on the informative (observed) sample, Bayesian prediction, predictive density, and predictive intervals are derived for future observation and decision. The performance of proposed methods are evaluated though a Monte Carlo simulation experiment. Two real-life datasets related to tax revenue and heath are incorporated to show the practical utility of proposed methodology in real phenomenon.

Statistical Analysis and Theoretical Framework for a Partially Accelerated Life Test Model with Progressive First Failure Censoring Utilizing a Power Hazard Distribution

Monitoring life-testing trials for a product or substance often demands significant time and effort. To expedite this process, sometimes units are subjected to more severe conditions in what is known as accelerated life tests. This paper is dedicated to addressing the challenge of estimating the power hazard distribution, both in terms of point and interval estimations, during constant- stress partially accelerated life tests using progressive first failure censored samples. Three techniques are employed for this purpose: maximum likelihood, two parametric bootstraps, and Bayesian methods. These techniques yield point estimates for unknown parameters and the acceleration factor. Additionally, we construct approximate confidence intervals and highest posterior density credible intervals for both the parameters and acceleration factor. The former relies on the asymptotic distribution of maximum likelihood estimators, while the latter employs the Markov chain Monte Carlo technique and focuses on the squared error loss function. To assess the effectiveness of these estimation methods and compare the performance of their respective confidence intervals, a simulation study is conducted. Finally, we validate these inference techniques using real-life engineering data.

Analysis of the Stress–Strength Model Using Uniform Truncated Negative Binomial Distribution under Progressive Type-II Censoring

In this study, we introduce a novel estimation technique for assessing the reliability parameter R=P(Y<X) of the uniform truncated negative binomial distribution (UTNBD) in the context of stress–strength analysis. We base our inferences on the assumption that both the strength (X) and stress (Y) random variables follow a UTNBD with identical first shape and scale parameters. In the presence of a progressive type-II censoring scheme, we employ maximum likelihood, two parametric bootstrap methods, and Bayesian estimation approaches to derive the estimators. Due to the complexity introduced by censoring, the estimators are not available in explicit forms and are instead obtained through numerical approximation techniques. Furthermore, we compute the highest posterior density credible intervals and determine the asymptotic variance-covariance matrix. To assess the performance of our proposed estimators, we conduct a Monte Carlo simulation study and provide a comparative analysis. Finally, we illustrate the practical applicability of our study with an engineering application.

Parametric inferences using dependent competing risks data with partially observed failure causes from MOBK distribution under unified hybrid censoring

In this communication, various statistical inferential procedures for estimating unknown model parameters are investigated via utilizing partially observed dependent competing risks data under the unified hybrid censoring scheme when the latent failure times follow Marshall–Olkin bivariate Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimators (MLEs) have been established. By using asymptotic normality property of MLE, the approximate confidence intervals have been constructed via observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals have been computed under a highly flexible gamma-Dirichlet prior distribution by using Markov chain Monte Carlo technique. In addition, to compare the performance of proposed methods, a Monte Carlo simulation has been carried out. Finally, a real-life data set has been analysed to illustrate the operability and applicability of the methods considered.

Estimating the real burden of gestational syphilis in Brazil, 2007–2018: a Bayesian modeling study

Although several studies have estimated gestational syphilis (GS) incidence in several countries, underreporting correction is rarely considered. This study aimed to estimate the level of under-registration and correct the GS incidence rates in the 557 Brazilian microregions. Brazilian GS notifications between 2007 and 2018 were obtained from the SINAN-Syphilis system. A cluster analysis was performed to group microregions according to the quality of GS notification. A Bayesian hierarchical Poisson regression model was applied to estimate the reporting probabilities among the clusters and to correct the associated incidence rates. We estimate that 45,196 (90%-HPD: 13,299; 79,310) GS cases were underreported in Brazil from 2007 to 2018, representing a coverage of 87.12% (90%-HPD: 79.40%; 95.83%) of registered cases, where HPD stands for the Bayesian highest posterior density credible interval. Underreporting levels differ across the country, with microregions in North and Northeast regions presenting the highest percentage of missed cases. After underreporting correction, Brazil's estimated GS incidence rate increased from 8.74 to 10.02 per 1000 live births in the same period. Our findings highlight disparities in the registration level and incidence rate of GS in Brazil, reflecting regional heterogeneity in the quality of syphilis surveillance, access to prenatal care, and childbirth assistance services. This study provides robust evidence to enhance national surveillance systems, guide specific policies for GS detection disease control, and potentially mitigate the harmful consequences of mother-to-child transmission. The methodology might be applied in other regions to correct disease underreporting. National Council for Scientific and Technological Development; The Bill Melinda Gates Foundation and Wellcome Trust.

Optimal sampling and statistical inferences for Kumaraswamy distribution under progressive Type-II censoring schemes

In this paper, we study non-Bayesian and Bayesian estimation of parameters for the Kumaraswamy distribution based on progressive Type-II censoring. First, the maximum likelihood estimates and maximum product spacings are derived. In addition, we derive the asymptotic distribution of the parameters and the asymptotic confidence intervals. Second, Bayesian estimators under symmetric and asymmetric loss functions (Squared error, linear exponential, and general entropy loss functions) are also obtained. The Lindley approximation and the Markov chain Monte Carlo method are used to derive the Bayesian estimates. Furthermore, we derive the highest posterior density credible intervals of the parameters. We further present an optimal progressive censoring scheme among different competing censoring scheme using three optimality criteria. Simulation studies are conducted to evaluate the performance of the point and interval estimators. Finally, one application of real data sets is provided to illustrate the proposed procedures.

Analysis of Muth parameters using generalized progressive hybrid censoring with application to sodium sulfur battery

This study takes into account the estimation concerns of the model parameter, reliability, and hazard rate functions of the Muth distribution when a sample is produced using generalized progressive hybrid censoring. The generalized progressive hybrid censoring scheme is suggested to get around the progressive hybrid censoring scheme's potential drawback of having extremely few failures. The maximum likelihood estimates of the model parameter and some life indices are obtained. The approximate confidence intervals for various parameters are calculated using the asymptotic distribution of the maximum likelihood estimates across the observed Fisher information matrix. Furthermore, the beta prior and squared error loss function are used to offer Bayes point estimates and the highest posterior density credible intervals of the unknown parameters, and the Markov Chain Monte Carlo technique is used to acquire the necessary estimates. We provide thorough simulation results to demonstrate the utility of the suggested methodologies. Finally, one genuine data set is analyzed as an example.

Estimation for partially observed left truncation and right censored competing risks data from a generalized inverted exponential distribution with illustrations

ABSTRACT In this paper, studies of competing risks model are considered when the observations are left-truncated and right-censored data. When the failure times of the competing risks are distributed by a generalized inverted exponential model with same scale but different shape parameters with partially observed failure causes, statistical inference for the unknown model parameters is discussed from classical and Bayesian approaches, respectively. Maximum likelihood estimators of the unknown parameters, along with associated existence and uniqueness, are established, and the asymptotic likelihood theory is also used to construct the confidence interval via the observed Fisher information matrix. Moreover, Bayesian estimates and the corresponding highest posterior density credible intervals are also obtained based a flexible Gamma-Beta prior, and a Gibbs sampling technique is constructed to compute associated estimates. Further, under a general practical assumption with order-restriction parameter case, classical and Bayesian estimations are also established under order restriction situations, respectively. Extensive Monte-Carlo simulations are carried out to investigate the performances of our results and two real-life examples are analyzed to show the applicability of the proposed methods.

Statistical Analysis of Inverse Lindley Data Using Adaptive Type-II Progressively Hybrid Censoring with Applications

This paper deals with the statistical inference of the unknown parameter and some life parameters of inverse Lindley distribution under the assumption that the data are adaptive Type-II progressively censored. The maximum likelihood method is considered to acquire the point and interval estimates of the distribution parameter, reliability, and hazard rate functions. The approximate confidence intervals are also addressed. The delta method is taken into consideration to approximate the variances of the estimators of the reliability and hazard rate functions to get the required intervals. Based on the assumption of gamma prior, we further consider Bayesian estimation of the different parameters. The Bayes estimates are obtained by considering squared error and general entropy loss functions. The Bayes estimates and highest posterior density credible intervals are obtained by employing the Markov chain Monte Carlo procedure. An exhaustive numerical study is conducted to compare the offered estimates with regard to their root means squared error, relative absolute biases, confidence lengths, and coverage probabilities. To explain the suggested methods, two applications are investigated. The numerical findings show that the Bayes estimates perform better than those obtained based on the maximum likelihood method. The Bayesian estimations using the asymmetric loss function give more efficient estimates than the symmetric loss. Finally, the inverse Lindley distribution is recommended to be used as a suitable model to fit airborne communication transceiver and wooden toys data sets when compared with some competitive models including inverse Weibull, inverse gamma and alpha power inverted exponential.