In this article, based on a general progressively Type-II censored data from the Pareto distribution of the second kind, we consider Bayesian and classical estimations of the parameters as well as some survival time parameters, the reliability, and hazard functions. Recently, Howlader and Hossain ( 2002 Howlader , H. A. , Hossain , A. ( 2002 ). Bayesian survival estimation of Pareto distribution of the second kind based on failure-censored data . Computat. Statist. Data Anal. 38 : 301 – 314 . [Crossref], [Web of Science ®] [Google Scholar] ) obtained Bayesian survival estimation using some approximation forms based on: quadratic loss function, non informative prior, and usually Type-II censored data. In this article, we propose to carry out a Bayesian estimation procedures based on: (i) informative prior for the shape parameter and discrete prior for the scale parameter of the model; (ii) both the symmetric loss function (squared error loss) and asymmetric loss function (linear-exponential (LINEX)) loss function; (iii) general progressively Type-II censored samples which include as a special cases: the regular doubly Type-II censored, progressively Type-II censored, usual Type-II right censored, and uncensored (complete) samples. The resulting Bayes estimates are obtained in closed form. Approximate confidence intervals for the parameters are constructed. Maximum likelihood estimators and their asymptotic variances are derived. One real data set has been analyzed for illustrative purposes. Finally, simulation study concerning different sample sizes and progressive censoring schemes were reported.

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.

In this article, based on progressively Type-II censored samples from a heterogeneous population that can be represented by a finite mixture of two-component Rayleigh lifetime model, the problem of estimating the parameters and some lifetime parameters (reliability and hazard functions) are considered. Both Bayesian and maximum likelihood estimators are of interest. A class of natural conjugate prior densities is considered in the Bayesian setting. The Bayes estimators are obtained using both the symmetric (squared error) loss function, and the asymmetric (LINEX and General Entropy) loss functions. It has been seen that the estimators obtained can be easily evaluated for this type of censoring by using suitable numerical methods. Finally, the performance of the estimates have been compared on the basis of their simulated maximum square error via a Monte Carlo simulation study.

The coefficient of variation (CV) is extensively used in many areas of applied statistics including quality control and sampling. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. In this article, based on progressive first-failure-censored data, we study the behavior of the CV of a random variable that follows a Burr-XII distribution. Specifically, we compute the maximum likelihood estimations and the confidence intervals of CV based on the observed Fisher information matrix using asymptotic distribution of the maximum likelihood estimator and also by using the bootstrapping technique. In addition, we propose to apply Markov Chain Monte Carlo techniques to tackle this problem, which allows us to construct the credible intervals. A numerical example based on real data is presented to illustrate the implementation of the proposed procedure. Finally, Monte Carlo simulations are performed to observe the behavior of the proposed methods.