Analyzing the dynamic system model with discrete failure time distribution

This article discusses the problem of estimation with step stress partially accelerated life tests using Type-II progressively censored samples. The lifetime of items under use condition follows the two-parameters inverted Kumaraswamy distribution. The maximum likelihood estimates for the unknown parameters are computed numerically. Using the property of asymptotic distributions for maximum likelihood estimation, we constructed asymptotic interval estimates. The Bayes procedure is used to calculate estimates of the unknown parameters from symmetrical and asymmetric loss functions. The Bayes estimates cannot be obtained explicitly, therefor the Lindley's approximation and the Markov chain Monte Carlo technique are used to obtaining the Bayes estimates. Furthermore, the highest posterior density credible intervals for the unknown parameters are calculated. An example is presented to illustrate the methods of inference. Finally, a numerical example of March precipitation (in inches) in Minneapolis failure times in the real world is provided to illustrate how the approaches will perform in practice.

In this work the analysis of interval-censored data, with Weibull distribution as the underlying lifetime distribution has been considered. It is assumed that censoring mechanism is independent and non-informative. As expected, the maximum likelihood estimators cannot be obtained in closed form. In our simulation experiments it is observed that the Newton-Raphson method may not converge many times. An expectation maximization algorithm has been suggested to compute the maximum likelihood estimators, and it converges almost all the times. The Bayes estimates of the unknown parameters under gamma priors are considered. If the shape parameter is known, the Bayes estimate of the scale parameter can be obtained in explicit form. When both the parameters are unknown, the Bayes estimators cannot be obtained in explicit form. Lindley’s approximation, importance sampling procedures and Metropolis Hastings algorithm are used to compute the Bayes estimates. Highest posterior density credible intervals of the unknown parameter are obtained using importance sampling technique. Small simulation experiments are conducted to investigate the finite sample performance of the proposed estimators, and the analysis of two data sets; one simulated and one real life, have been provided for illustrative purposes.

In this paper, the statistical inference of the unknown parameters of a Burr Type III (BIII) distribution based on the unified hybrid censored sample is studied. The maximum likelihood estimators of the unknown parameters are obtained using the Expectation–Maximization algorithm. It is observed that the Bayes estimators cannot be obtained in explicit forms, hence Lindley's approximation and the Markov Chain Monte Carlo (MCMC) technique are used to compute the Bayes estimators. Further the highest posterior density credible intervals of the unknown parameters based on the MCMC samples are provided. The new model selection test is developed in discriminating between two competing models under unified hybrid censoring scheme. Finally, the potentiality of the BIII distribution to analyze the real data is illustrated by using the fracture toughness data of the three different materials namely silicon nitride (Si3N4), Zirconium dioxide (ZrO2) and sialon (Si6−xAlxOxN8−x). It is observed that for the present data sets, the BIII distribution has the better fit than the Weibull distribution which is frequently used in the fracture toughness data analysis.

The study deals with the analysis of Type-II hybrid censored data from the modified Weibull distribution. We provide maximum-likelihood estimates of the parameters, reliability, and hazard rate functions along with their standard errors. The confidence intervals along with their widths have also been obtained. Assuming gamma and Jeffrey’s invariant priors for the unknown parameters, Bayes estimates along with its posterior errors and highest posterior density credible intervals are obtained. The Markov Chain Monte Carlo technique has been used to simulate draws from the complicated posterior densities of the parameters. A simulation study is conducted to compare the performances of classical and Bayesian methods of estimation. Finally, a real data analysis is performed for illustrative purpose.

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.

A hybrid censoring scheme is a mixture of type-I and type-II censoring schemes. This article presents the statistical inferences on Weibull parameters when the data are type-II hybrid censored. The maximum likelihood estimators, and the approximate maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the maximum likelihood estimators are used to construct approximate confidence intervals. Bayes estimates, and the corresponding highest posterior density credible intervals of the unknown parameters, are obtained using suitable priors on the unknown parameters, and by using Markov chain Monte Carlo techniques. The method of obtaining the optimum censoring scheme based on the maximum information measure is also developed. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyse one data set for illustrative purposes.

Using progressive Type-II censoring data, this study deals with the estimation of parameters of the Unit-Weibull distribution using two classical methods and the Bayesian method. In the classical methods, maximum likelihood and the maximum product of spacing (MPS) methods are used to obtain the parameters of the model by utilizing the Newton–Raphson method. On the basis of observed Fisher information matrix, approximate confidence intervals for the unknown parameters are obtained. In addition, two bootstrap methods are used to obtain confidence intervals for the unknown parameters of the model. In the Bayesian estimation, we have considered both likelihood function as well as product of spacing function to estimate the model parameters. Bayes estimators are obtained under squared error loss function using independent gamma density priors for the unknown model parameters. Since closed-form of the Bayes estimators is not available, the Metropolis–Hastings algorithm is proposed to approximate the Bayes estimates. In addition, highest posterior density credible intervals are obtained. Further, using different optimally criteria, an optimal scheme has been proposed. A simulation study is conducted to assess the statistical performance of all the estimators. To demonstrate the proposed methodology a real data analysis is provided to illustrate all the statistical inferential procedures developed in the paper.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Assessing the reliability of two production lines, whether individually or simultaneously, is of significant importance for improving manufacturing processes and guaranteeing superior product quality. This paper examines the reliability assessment of two production lines utilizing joint progressively Type-II censored samples derived from the XLindley distribution. In addition to estimating the unknown parameters, the reliability functions for each production line, as well as for both production lines simultaneously, are analyzed. Both classical likelihood-based and Bayesian methodologies are employed for estimation purposes. The maximum likelihood method is applied to obtain point estimates for the unknown parameters and reliability functions, while Bayesian analysis is performed under the squared error loss function, employing the Markov Chain Monte Carlo technique to generate samples from the posterior distribution. The approximate confidence intervals, percentile bootstrap confidence intervals, and the highest posterior density credible intervals for the unknown parameters and various reliability functions are discussed. The problem of selecting the optimal censoring plan is also considered. A comprehensive simulation study is conducted to assess the performance of the proposed methods, comparing the accuracy and efficiency of the estimates across different censoring schemes. Furthermore, the applicability of the proposed methodology is demonstrated through the analysis of two real-world data sets, underscoring its practical utility within the field of reliability. Finally, three criteria, namely A-optimality, D-optimality, and F-optimality, are considered to determine the optimal censoring plan.

Bayesian inference of Weibull distribution based on left truncated and right censored data

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.

The unified hybrid censoring is a mixture of generalized Type I and Type II hybrid censoring schemes. This paper presents the statistical inference and prediction of the Burr Type XII distribution for unified hybrid-censored data. The maximum likelihood estimators for estimating the unknown parameters are developed using the expectation-maximization algorithm. The Bayesian estimates of the unknown parameters under the assumption of independent gamma priors are obtained using two approximations, namely Lindley's approximation and the Markov Chain Monte Carlo (MCMC) technique. Also, MCMC samples are used to construct the highest posterior density credible intervals as well. Simulation experiments are performed to compare the different proposed methods. The prediction of the future observations based on a unified hybrid-censored sample is also provided. A real dataset representing the droplet splashing reported in 90° spray angle is used to illustrate the derived results. Furthermore, a useful interval say tracking interval is introduced for comparing the distance of the two rival models from the true model.

As an extension of the standard Weibull distribution, a new crucial distribution termed alpha power Weibull distribution has been presented. It can model decreasing, increasing, bathtub, and upside-down bathtub failure rates. This research investigates the estimation of model parameters and some of its reliability characteristics using progressively Type-II censored data. To get estimates of unknown parameters, reliability, and hazard rate functions, the maximum likelihood, and Bayesian estimation approaches are studied. To acquire estimated confidence intervals for unknown parameters and reliability characteristics, the maximum likelihood asymptotic properties are used. The Markov chain Monte Carlo approach is used in Bayesian estimation to provide Bayesian estimates under squared error and LINEX loss functions. Furthermore, the highest posterior density credible intervals of the parameters and reliability characteristics are determined. A Monte Carlo simulation study is used to investigate the accuracy of various point and interval estimators. In addition, various optimality criteria are used to choose the best progressive censoring schemes. Two real data from the engineering field are analyzed to demonstrate the applicability and significance of the proposed approaches. Based on numerical results, the Bayesian procedure for estimating the parameters and reliability characteristics of alpha power Weibull distribution is recommended. The analysis of two real data sets showed that the alpha power Weibull distribution is a good model to investigate engineering data in the presence of progressive Type-II censoring.

In the sample survey, it is impossible to obtain sufficient samples in relatively small areas, which makes it difficult to estimate accurate statistics for small areas. For this reason, various studies on small area estimation are constantly being conducted, and in particular, the Fay-Herriot (1979) model has been expanded to various forms as one of the most widely used small area estimation models. In this study, we developed a semiparametric regression model based on a radial basis function that can consider nonlinear relationships between the resulting variables and covariates, extended from the model considering the measurement error of the covariate in the Poisson distribution proposed by Nam, Hwang(2021). Also, parameter estimation and model fit were performed through hierarchical Bayesian estimation using Gibbs sampling and Metropolis-Hastings algorithm among Markov chain Monte Carlo techniques. Simulations were conducted under various conditions to confirm the suitability of the model developed in this study, and the excellence of the developed model was further verified through empirical analysis using data from the 8th Korean longitudinal study of aging.

The ranked set sampling (RSS) is an efficient and flexible sampling method. Based on a modified RSS named minimum ranked set sampling samples (MinRSSU), inference of a dependent competing risks model is proposed in this paper. Then, Marshall–Olkin bivariate distribution model is used to describe the dependence of competing risks. When the competing risks data follow the proportional hazard rate distribution, a dependent competing risks model based on MinRSSU sampling is constructed. In addition, the model parameters and reliability indices were estimated by the classical and Bayesian method. Maximum likelihood estimators and corresponding asymptotic confidence intervals are constructed by using asymptotic theory. In addition, the Bayesian estimator and highest posterior density credible intervals are established under the general prior. Furthermore, according to E-Bayesian theory, the point and interval estimators of model parameters and reliability indices are obtained by a sampling algorithm. Finally, extensive simulation studies and a real-life example are presented for illustrations.