Estimation and Prediction for a Progressively First-Failure Censored Inverted Exponentiated Rayleigh Distribution

This article deals with the estimation of parameters and reliability characteristics of Lindley distribution underrandom censoring. Expected time on test based on randomly censored data is obtained. The maximum likelihood estimators of the unknown parameters and reliability characteristics are derived. The asymptotic, bootstrap p and bootstrap t confidence intervals of the parameters are constructed. The Bayes estimators of the parameters and reliability characteristics under squared error loss function using non-informative and gamma informative priors are obtained. For computing of Bayes estimates, Lindley approximation and MCMC methods are considered. Highest posterior density (HPD) credible intervals of the parameters are obtained using MCMC method. Various estimation procedures are compared using a Monte Carlo simulation study. Finally, a real data set is analyzed for illustration purposes.

The Burr-XII distribution has been widely applied in engineering, reliability, and survival analysis. Due to its importance, in this article, statistical inferences for Burr-XII distribution under a joint type-II censoring scheme is discussed. The classical likelihood estimation of unknown model parameters is studied via different calculating approaches, such as the expectation-maximization (EM) algorithm and approximate confidence intervals (ACIs) using the observed Fisher information matrix are obtained. The asymptotic bootstrap confidence intervals are also computed. In the sequel, Bayesian estimations of unknown parameters with a gamma prior distribution are considered under squared error, linear-exponential, and generalized entropy loss functions. Subsequently, we calculate the Bayesian credible interval using the importance sample. The performance of the developed methods is investigated through a Monte Carlo simulation study and two real-life examples. The results showed that the proposed estimation strategies have satisfactory results. However, Bayesian approaches were preferable to EM in terms of lower mean square error and higher coverage probability.

This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape parameters. First, the Newton-Raphson method is used to derive the maximum likelihood estimators (MLEs), and the existence and uniqueness of the estimators are also demonstrated. We used the stochastic expectation maximization (SEM) method to construct MLEs for unknown parameters, which simplified and facilitated computation. Based on the asymptotic normality of the MLEs and SEM methods, we create the corresponding confidence intervals for unknown parameters, and the delta approach is utilized to obtain the interval estimation of the reliability function. Additionally, using two bootstrap techniques, the approximative interval estimators for all unknowns are created. Furthermore, we computed the Bayes estimates of unknown parameters as well as the survival function using the Markov chain Monte Carlo (MCMC) method in the presence of square error and LINEX loss functions. Finally, we look into two real data sets and create a simulation study to evaluate the efficacy of the established approaches.

Let a progressively type-II (PT-II) censored sample of size m is available. Under this set-up, we consider the problem of estimating unknown model parameters and two reliability characteristics of the log-logistic distribution. Maximum likelihood estimates (MLEs) are obtained. We use expectation–maximization (EM) algorithm. The observed Fisher information matrix is computed. We propose Bayes estimates with respect to various loss functions. In this purpose, we adopt Lindley’s approximation and importance sampling methods. Asymptotic and bootstrap confidence intervals are derived. Asymptotic intervals are obtained using two approaches: normal approximation to MLEs and log-transformed MLEs. The bootstrap intervals are computed using boot-t and boot-p algorithms. Further, highest posterior density (HPD) credible intervals are constructed. Two sets of practical data are analyzed for the illustration purpose. Finally, detailed simulation study is carried out to observe the performance of the proposed methods.

The analysis of lifetime data under censoring schemes plays a vital role in reliability studies and survival analysis, where complete information is often difficult to obtain. This work focuses on the estimation of the parameters of the recently proposed generalized Kavya–Manoharan exponential (GKME) distribution under progressive Type-I interval censoring, a censoring scheme that frequently arises in medical and industrial life-testing experiments. Estimation procedures are developed under both classical and Bayesian paradigms, providing a comprehensive framework for inference. In the Bayesian setting, parameter estimation is carried out using Markov Chain Monte Carlo (MCMC) techniques under two distinct loss functions: the squared error loss function (SELF) and the general entropy loss function (GELF). For interval estimation, asymptotic confidence intervals as well as highest posterior density (HPD) credible intervals are constructed. The performance of the proposed estimators is systematically evaluated through a Monte Carlo simulation study in terms of mean squared error (MSE) and the average lengths of the interval estimates. The practical usefulness of the developed methodology is further demonstrated through the analysis of a real dataset on survival times of guinea pigs exposed to virulent tubercle bacilli. The findings indicate that the proposed methods provide flexible and efficient tools for analyzing progressively interval-censored lifetime data.

We consider estimation of the multicomponent stress-strength reliability under progressive Type II censoring under the assumption that stress and strength variables follow Burr XII distributions with a common shape parameter. Maximum likelihood estimates of the reliability are obtained along with asymptotic intervals when common shape parameter may be known or unknown. Bayes estimates are also derived under the squared error loss function using different approximation methods. Further, we obtain exact Bayes and uniformly minimum variance unbiased estimates of the reliability for the case common shape parameter is known. The highest posterior density intervals are also obtained. We perform Monte Carlo simulations to compare the performance of proposed estimates and present a discussion based on this study. Finally, two real data sets are analyzed for illustration purposes.

In this article analysis of a simple step‐stress accelerated life test is considered under progressive type‐II censoring. A cumulative exposure model is considered when the latent lifetimes of test units follow the Gompertz distribution with different shape parameters and a common scale parameter. We explore the study by estimating all unknown parameters using classical and Bayesian techniques. The model parameters are estimated using maximum likelihood and Bayesian methods. Subsequently, interval estimates are derived based on the observed Fisher information matrix. Bayesian estimates are obtained using squared error and linear exponential loss functions. Subsequently highest posterior density intervals are also constructed. We examine the efficiency of all estimators through simulation studies. Finally, we provide a real‐life example in support of the considered model.

ABSTRACTIn this article, we consider the multiple step stress model based on the cumulative exposure model assumption. Here, it is assumed that for a given stress level, the lifetime of the experimental units follows exponential distribution and the expected lifetime decreases as the stress level increases. We mainly focus on the order restricted inference of the unknown parameters of the lifetime distributions. First we consider the order restricted maximum likelihood estimators (MLEs) of the model parameters. It is well known that the order restricted MLEs cannot be obtained in explicit forms. We propose an algorithm that stops in finite number of steps and it provides the MLEs. We further consider the Bayes estimates and the associated credible intervals under the squared error loss function. Due to the absence of explicit form of the Bayes estimates, we propose to use the importance sampling technique to compute Bayes estimates. We provide an extensive simulation study in case of three stress levels mainly to see the performance of the proposed methods. Finally the analysis of one real data set has been provided for illustrative purposes.

The progressively first failure censored (PFFC) data have become very popular in the past decade due to its usefulness in life testing experiments and in reliability theory. The PFFC data record the number of failures and increase the efficiency of the estimators. The inverse Pareto distribution (IPD) is useful when empirical data suggest a decreasing or upside-down bathtub-shaped failure rate functions. In this article, we consider the classical and Bayesian estimation of the model parameter and the reliability characteristics of the IPD using the PFFC data. The maximum likelihood estimators, asymptotic confidence, and bootstrap confidence intervals are considered in the classical estimation. Under the Bayesian paradigm, Bayes estimators based on non-informative and the gamma informative priors under the squared error loss function using Tierney-Kadane approximation, importance sampling, and Metropolis-Hasting (M-H) algorithm are assessed. In addition, the highest probability density (HPD) intervals based on the M-H algorithm are also constructed. To investigate the efficacy of each of the estimation procedures, numerical computations are performed based on a simulation study. Finally, a real data set is re-analyzed to show the applicability of the IPD model under a censoring scheme.

Reliability analysis is the key to evaluate software’s quality. Since the early 1970s, the Power Law Process, among others, has been used to assess the rate of change of software reliability as time-varying function by using its intensity function. The Bayesian analysis applicability to the Power Law Process is justified using real software failure times. The choice of a loss function is an important entity of the Bayesian settings. The analytical estimate of likelihood-based Bayesian reliability estimates of the Power Law Process under the squared error and Higgins-Tsokos loss functions were obtained for different prior knowledge of its key parameter. As a result of a simulation analysis and using real data, the Bayesian reliability estimate under the Higgins-Tsokos loss function not only is robust as the Bayesian reliability estimate under the squared error loss function but also performed better, where both are superior to the maximum likelihood reliability estimate. A sensitivity analysis resulted in the Bayesian estimate of the reliability function being sensitive to the prior, whether parametric or non-parametric, and to the loss function. An interactive user interface application was additionally developed using Wolfram language to compute and visualize the Bayesian and maximum likelihood estimates of the intensity and reliability functions of the Power Law Process for a given data.

Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.

ABSTRACTWe consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.

In this paper, we focus on the parametric inference based on the Tampered Random Variable (TRV) model for simple step-stress life testing (SSLT) using Type-II censored data. The baseline lifetime of the experimental units, under normal stress conditions, follows the Gumbel Type-II distribution with $\alpha$ and $\lambda$ being the shape and scale parameters, respectively. Maximum likelihood estimator (MLE) and Bayes estimator of the model parameters are derived based on Type-II censored samples. We obtain asymptotic intervals of the unknown parameters using the observed Fisher information matrix. Bayes estimators are obtained using Markov Chain Monte Carlo (MCMC) method under squared error loss and LINEX loss functions. We also construct highest posterior density (HPD) intervals of the unknown model parameters. Extensive simulation studies are performed to investigate the finite sample properties of the proposed estimators. Three different optimality criteria have been considered to determine the optimal censoring plans. Finally, the methods are illustrated with the analysis of two real data sets.

Coherent systems are widely studied in reliability experiments. Under the assumption that the components of a coherent system follow a two-parameter Weibull distribution, maximum likelihood inference for n-component coherent systems with known signatures under a simple step-stress model is discussed in this paper. The detailed steps of the stochastic expectation maximization algorithm under this setup are also developed to obtain estimates of the model parameters. Asymptotic confidence intervals for the model parameters are constructed using the observed Fisher information matrix and missing information principle. Parametric bootstrap approach is used also to construct confidence intervals for the parameters. A method based on best linear unbiased estimators is developed to provide initial values that are needed for numerical computation of maximum likelihood estimates. The performance of the methods developed is assessed through an extensive Monte Carlo simulation study. Finally, two numerical examples are presented for illustrative purpose.

Finite mixtures of multivariate skew t (MST) distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. A number of algorithms for the computation of the maximum likelihood (ML) estimates for the model parameters of mixtures of MST distributions have been put forward in recent years. These implementations use various characterizations of the MST distribution, which are similar but not identical. While exact implementation of the expectation-maximization (EM) algorithm can be achieved for `restricted' characterizations of the component skew t-distributions, Monte Carlo (MC) methods have been used to fit the `unrestricted' models. In this paper, we review several recent fitting algorithms for finite mixtures of multivariate skew t-distributions, at the same time clarifying some of the connections between the various existing proposals. In particular, recent results have shown that the EM algorithm can be implemented exactly for faster computation of ML estimates for mixtures with unrestricted MST components. The gain in computational time is effected by noting that the semi-infinite integrals on the E-step of the EM algorithm can be put in the form of moments of the truncated multivariate non-central t-distribution, similar to the restricted case, which subsequently can be expressed in terms of the non-truncated form of the central t-distribution function for which fast algorithms are available. We present comparisons to illustrate the relative performance of the restricted and unrestricted models, and demonstrate the usefulness of the recently proposed methodology for the unrestricted MST mixture, by some applications to three real datasets.