E-Bayesian Estimation of some lifetime distributions and Hierarchical models

Abstract

This study innovates the idea of E-Bayesian theory using different lifetime distributions in order to assess the performance of various loss functions. In addition, we propose Hierarchical models to estimate its unknown parameter(s) using E-Bayesian theory. Moreover, we introduce Progressive Type-II censoring scheme in this research. The significant shortcoming of conventional Type-I & II censoring schemes is that the units under experimentation may be dismissed only when the experiment terminates. But a situation may occurs when infor mation with units under observation are lost due to certain reasons and such reasons lead theorists into the domain of progressive censoring scheme. The E-Posterior Risk, MSE and E-MSE have been used as evaluation criteria. The Monte Carlo simulations are performed to analyze the efficiency of E-Bayesian estimators through empirical observation and the associated real life examples are also studied. To begin with, we apply E-Bayesian estimation to estimate the rate parameter of Maxwell distribution using various loss functions under uniform hyper-prior. In addition, we introduce the idea of empirical E-Bayesian estimation together with Hierarchical modelling and propose E-Bayesian estimation to estimate the parameter of the Hierarchical Poisson-Gamma model under scaled square error loss function. Moreover, we coalesce the bathtub-shaped lifetime distribution with progressive Type-II censoring scheme and the aim is to estimate the scale parameter under restricted parameter space using Bayesian and E-Bayesian estimations on the basis of uniform hyper-prior. It is found that the E-Bayesian estimation is more robust than Bayesian and Classical methods of estimation. Finally, we initiate the idea of Hierarchical modelling to derive the variance parameter of Hierarchical Normal and inverse gamma model using E-Bayesian estimation under three distinct bi-variate independent hyper-priors. We propose the idea of a Hierarchical probability density function instead of the traditional Hierarchical prior density function. The asymptotic properties of the E-Bayesian estimators are also evaluated. The Monte Carlo simulations are operated to assess the precision of proposed estimators along with real life applications and it shows that E-Bayesian estimation is better to operate.