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The estimations of parameters and percentiles for a three‐parameter Marshall Olkin extended inverse Weibull distribution based on progressively type‐II censored sample are concerned. The Bayesian, least‐squares, maximum likelihood and percentiles estimate methods for the model parameters have been developed. Comparing all estimate methods, the least‐squares, maximum likelihood and percentiles estimate methods are shown not stable due to the identification problem in the extended parametric space. Therefore, the Bayes estimate methods are focused. Three Bayesian estimations of the distribution parameters and p percentiles for p=$p =$ 2.5, 50 and 97.5 under the squared error loss, absolute error loss and LINEX loss functions are respectively calculated by using the Markov chain Monte Carlo sampling procedure. Moreover, two real data sets are presented for the application illustration. Finally, concluding remarks are addressed. [ABSTRACT FROM AUTHOR]

Given a sample, E-Bayesian estimates, which are the expected Bayesian estimators over the joint distributions of two hyperparameters in the prior distribution, are developed for the inverse Weibull distribution rate parameter under the scaled squared error and linear exponential error loss functions, respectively. The corresponding expected mean square errors, EMSEs, of E-Bayesian estimators based on the sample are derived. Moreover, the theoretical properties of EMSEs are established. A Monte Carlo simulation study is conducted for the performance comparison. Finally, three data sets are given for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we consider Dagum distribution which is capable of modeling various shapes of failure rates and aging criteria. Based on progressively type-I interval censoring data, we first obtain the maximum likelihood estimators and the approximate confidence intervals of the unknown parameters of the Dagum distribution. Next, we obtain the Bayes estimators of the parameters of Dagum distribution under the squared error loss (SEL) and balanced squared error loss (BSEL) functions using independent informative gamma and non informative uniform priors for both scale and two shape parameters. A Monte Carlo simulation study is performed to assess the performance of the proposed Bayes estimators with the maximum likelihood estimators. We also compute credible intervals and symmetric 100(1 − τ)% two-sided Bayes probability intervals under the respective approaches. Besides, based on observed samples, Bayes predictive estimates and intervals are obtained using one-and two-sample schemes. Simulation results reveal that the Bayes estimates based on SEL and BSEL performs better than maximum likelihood estimates in terms of bias and MSEs. Besides, credible intervals have smaller interval lengths than confidence interval. Further, predictive estimates based on SEL with informative prior performs better than non-informative prior for both one and two sample schemes. Further, the optimal censoring scheme has been suggested using a optimality criteria. Finally, we analyze a data set to illustrate the results derived. [ABSTRACT FROM AUTHOR]

Type-II censored data is an important scheme of data in lifetime studies. The purpose of this paper is to obtain E-Bayesian predictive functions which are based on observed order statistics with two samples from two parameter Burr XII model. Predictive functions are developed to derive both point prediction and interval prediction based on type-II censored data, where the median Bayesian estimation is a novel formulation to get Bayesian sample prediction, as the integral for calculating the Bayesian prediction directly does not exist. All kinds of predictions are obtained with symmetric and asymmetric loss functions. Two sample techniques are considered, and gamma conjugate prior density is assumed. Illustrative examples are provided for all the scenarios considered in this article. Both illustrative examples with real data and the Monte Carlo simulation are carried out to show the new method is acceptable. The results show that Bayesian and E-Bayesian predictions with the two kinds of loss functions have little difference for the point prediction, and E-Bayesian confidence interval (CI) with the two kinds of loss functions are almost similar and they are more accurate for the interval prediction. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with using the E-Bayesian method [M. Han, Applied Mathematical Modeling (2009) 1915–1922] for computing estimates for the parameter and reliability function of the Burr type XII distribution based on type-2 censored samples. The estimates are obtained based on squared error and LINEX loss functions. A comparison between the new method and the corresponding Bayes and maximum likelihood techniques is made using the Monte Carlo simulation. [Copyright &y& Elsevier]

The recent exponentiated generalized linear exponential distribution is a generalization of the generalized linear exponential distribution and the exponentiated generalized linear exponential distribution. In this paper, we study some statistical properties of this distribution such as negative moments, moments of order statistics, mean residual lifetime, and their asymptotic distributions for sample extreme order statistics. Different estimation procedures include the maximum likelihood estimation, the corrected maximum likelihood estimation, the modified maximum likelihood estimation, the maximum product of spacing estimation, and the least squares estimation are compared via a Monte Carlo simulation study in terms of their biases, mean squared errors, and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate its use for modeling a rainfall data from Orlando, Florida. [ABSTRACT FROM AUTHOR]

Bayesian estimates involve the selection of hyper-parameters in the prior distribution. To deal with this issue, the empirical Bayesian and E-Bayesian estimates may be used to overcome this problem. The first one uses the maximum likelihood estimate (MLE) procedure to decide the hyper-parameters; while the second one uses the expectation of the Bayesian estimate taken over the joint prior distribution of the hyper-parameters. This study focuses on establishing the E-Bayesian estimates for the Lomax distribution shape parameter functions by utilizing the Gamma prior of the unknown shape parameter along with three distinctive joint priors of Gamma hyper-parameters based on the square error as well as two asymmetric loss functions. These two asymmetric loss functions include a general entropy and LINEX loss functions. To investigate the effect of the hyper-parameters' selections, mathematical propositions have been derived for the E-Bayesian estimates of the three shape functions that comprise the identity, reliability and hazard rate functions. Monte Carlo simulation has been performed to compare nine E-Bayesian, three empirical Bayesian and Bayesian estimates and MLEs for any aforementioned functions. Additionally, one simulated and two real data sets from industry life test and medical study are applied for the illustrative purpose. Concluding notes are provided at the end. [ABSTRACT FROM AUTHOR]