The Weibull Claim Model: Bivariate Extension, Bayesian, and Maximum Likelihood Estimations

This study introduces a robust Bayesian framework for simple-step stress accelerated life testing. For modeling lifetime characteristics, this study adopts the Xgamma distribution as the principal life model. Bayesian parameter estimation is conducted using the Markov Chain Monte Carlo method under three distinct loss functions: the squared error loss function, generalized entropy loss function, and linear exponential loss function. The resulting Bayesian estimators are analyzed and compared with those obtained through the classical maximum likelihood estimation method. Finally, the proposed methods are illustrated with the analysis of two real data sets.

This study addresses the challenge of estimating parameters for two logistic populations that share a common scale parameter but have different location parameters in the presence of fuzzy data. To handle these complexities, both Maximum Likelihood Estimation (MLE) and Bayesian methods are employed. Asymptotic confidence intervals are constructed using ML estimates. For Bayesian estimation, a conjugate prior is utilized, and Bayes estimators are approximated using Lindley’s method due to the lack of closed-form solutions. Furthermore, Approximate Bayesian Computation (ABC) and Markov Chain Monte Carlo (MCMC) techniques, including Hamiltonian Monte Carlo (HMC) and the Metropolis–Hastings (MH) algorithm, are utilized to sample from the posterior distributions and construct Highest Posterior Density (HPD) intervals. A detailed comparative analysis of MLE, Lindley’s approximation, ABC, HMC, and MH is conducted to assess their performance. The effectiveness of the proposed methodology is demonstrated using a real-world dataset under fuzzy conditions.

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.

In this paper, we propose maximum likelihood estimators and Bayes estimators of parameters of exponentiated Pareto distribution under general Entropy loss function and squared error loss function for Progressive type-II censored data with binomial removals. The maximum likelihood estimators and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from exponentiated Pareto distribution.

In this article, we considered the problem of estimation for the parameters and reliability characteristics of Marshall-Olkin Extended exponential distribution (MOEED) using hybrid Type-II censored data. For estimation, the maximum likelihood estimators (MLE’s) andBayes estimators are constructed. In Bayesian paradigm, Markov Chain Monte Carlo method has been used to compute the Bayes estimates. The 95% asymptotic confidence intervals and highest posterior density intervals for the parameters are also obtained. A comparative study between maximum likelihood estimators and Bayes estimators has been performed using simulation technique. Lastly, a real data set is also analysed for illustration.

In this article, we propose Maximum likelihood estimators (MLEs) and Bayes estimators of parameters of Poisson-exponential distribution (PED) under General entropy loss function (GELF) and Squared error loss function (SELF) for Progressive type-II censored data with binomial removals (PT-II CBRs). The MLEs and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from PED. The proposed methodology is illustrated on a real dataset of ovarian cancer.

It is known that various types of location privacy attacks can be carried out using a personalized transition matrix that is learned for each target user, or a population transition matrix that is common to all target users. However, since many users disclose only a small amount of location information in their daily lives, the training data can be extremely sparse. The aim of this paper is to clarify the risk of location privacy attacks in this realistic situation. To achieve this aim, we propose a learning method that uses tensor factorization (or matrix factorization) to accurately estimate personalized transition matrices (or a population transition matrix) from a small amount of training data. To avoid the difficulty in directly factorizing the personalized transition matrices (or population transition matrix), our learning method first factorizes a transition count tensor (or matrix), whose elements are the number of transition counts that the user has made, and then normalizes counts to probabilities. We focus on a localization attack, which derives an actual location of a user at a given time instant from an obfuscated trace, and compare our learning method with the maximum likelihood (ML) estimation method in both the personalized matrix mode and the population matrix mode. The experimental results using four real data sets show that the ML estimation method performs only as well as a random guess in many cases, while our learning method significantly outperforms the ML estimation method in all of the four data sets.

ABSTRACTThis article addresses the various properties and different methods of estimation of the unknown parameter of length and area-biased Maxwell distributions. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of length and area-biased Maxwell distributions (such as moments, moment-generating function (mgf), hazard rate function, mean residual lifetime function, residual lifetime function, reversed residual life function, conditional moments and conditional mgf, stochastic ordering, and measures of uncertainty) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimator, moments estimator, least-square and weighted least-square estimators, maximum product of spacings estimator and compare them using extensive numerical simulations. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using inverted gamma prior for the scale parameter. Furthermore, Bayes estimators and their respective posterior risks are computed and compared using Markov chain Monte Carlo (MCMC) algorithm. Also, bootstrap confidence intervals using frequentist approaches are provided to compare with Bayes credible intervals. Finally, a real dataset has been analyzed for illustrative purposes.

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

Progressive Type-II hybrid censored schemes based on maximum product spacing with application to Power Lomax distribution

<abstract><p>The Power Rayleigh distribution (PRD) is a new extension of the standard one-parameter Rayleigh distribution. To employ this distribution as a life model in the analysis of reliability and survival data, we focused on the statistical inference for the parameters of the PRD under the adaptive Type-II censored scheme. Point and interval estimates for the model parameters and the corresponding reliability function at a given time are obtained using likelihood, Bootstrap and Bayesian estimation methods. A simulation study is conducted in different settings of the life testing experiment to compare and evaluate the performance of the estimates obtained. In addition, the estimation procedure is also investigated in real lifetimes data. The results indicated that the obtained estimates gave an accurate and efficient estimation of the model parameters. The Bootstrap estimates are better than the estimates obtained by the likelihood estimation approach, and estimates obtained using the Markov Chain Monte Carlo method by the Bayesian approach under both the squared error and the general entropy loss functions have priority over other point and interval estimates. Under the adaptive Type-II censoring scheme, concluding results confirmed that the PRD can be effectively used to model the lifetimes in survival and reliability analysis.</p></abstract>

In this paper we develop approximate Bayes estimators of the parameters, reliability, and hazard rate functions of the Logistic distribution by using Lindley’s approximation, based on progressively type-II censoring samples. Noninformative prior distributions are used for the parameters. Quadratic, linex and general Entropy loss functions are used. The statistical performances of the Bayes estimates relative to quadratic, linex and general entropy loss functions are compared to those of the maximum likelihood based on simulation study.

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.

In this paper, some estimators for the reliability function R(t) of Basic Gompertz (BG) distribution have been obtained, such as Maximum likelihood estimator, and Bayesian estimators under General Entropy loss function by assuming non-informative prior by using Jefferys prior and informative prior represented by Gamma and inverted Levy priors. Monte-Carlo simulation is conducted to compare the performance of all estimates of the R(t), based on integrated mean squared.

This paper examines the estimation of parameters, reliability, and hazard function of the generalized inverted exponential distribution within an improved adaptive Type‐II progressive censoring framework. Point estimates are derived through frequentist and Bayesian methodologies employing the Expectation‐Maximization (EM) algorithm, Markov chain Monte Carlo (MCMC), and TK approximation, while interval estimates are established utilizing the asymptotic normality property of maximum likelihood estimates (MLE) and MCMC samples. A generalized entropy loss function is considered, and extensive simulations are conducted to compare the effectiveness of the proposed methodologies. Additionally, two real datasets are analyzed to illustrate the practical applicability of our study.