E-Bayesian estimations of parameter and its evaluation standard: E-MSE (expected mean square error) under different loss functions

Abstract This paper is concerned with using the E-Bayesian method for computing estimates of Pareto index. In order to measure the estimated error, in the case of the one hyper parameter, the definition of E-MSE (expected mean square error) is proposed based on the definition of E-Bayesian estimation. Moreover, the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively, these estimations are derived based on a conjugate prior distribution under different loss functions (including: squared error loss, weighted squared error loss and precautionary loss). Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation, results are compared on basis of E-MSE. Finally, combined with the golfers income problem are performed to calculated (also using OpenBUGS), and for income inequality degree were performed to compared and analyzed. When considering evaluating the E-Bayesian estimations under different loss functions, this paper proposed the E-MSE as evaluation standard.

[1]  Verónica Andrea González-López,et al.  E-Bayesian estimation for system reliability and availability analysis based on exponential distribution , 2017, Commun. Stat. Simul. Comput..

[2]  Ali Karimnezhad,et al.  Bayes, E-Bayes and robust Bayes prediction of a future observation under precautionary prediction loss functions with applications , 2016 .

[3]  Fatih Kızılaslan.  The E-Bayesian and hierarchical Bayesian estimations for the proportional reversed hazard rate model based on record values , 2017 .

[4]  Ming Han,et al.  E-Bayesian estimation and its E-MSE under the scaled squared error loss function, for exponential distribution as example , 2019, Commun. Stat. Simul. Comput..

[5]  زينهم فكري جاهين,et al.  E-Bayesian estimation for the Burr type XII model based on type-2 censoring , 2011 .

[6]  N. Nematollahi,et al.  Estimation of scale parameter under entropy loss function , 1996 .

[7]  Hassan M. Okasha,et al.  E-Bayesian estimation for the geometric model based on record statistics , 2016 .

[8]  M. Aslam,et al.  PREFERENCE OF PRIOR FOR THE CLASS OF LIFE-TIME DISTRIBUTIONS UNDER DIFFERENT LOSS FUNCTIONS , 2012 .

[9]  Adil Mousa Younis,et al.  Quasi-E-Bayesian criteria versus quasi-Bayesian, quasi-hierarchical Bayesian and quasi-empirical Bayesian methods for estimating the scale parameter of the Erlang distribution , 2016 .

[10]  Sajid Ali,et al.  On the Bayesian estimation of the weighted Lindley distribution , 2015 .

[11]  F. Yousefzadeh.  E-Bayesian and hierarchical Bayesian estimations for the system reliability parameter based on asymmetric loss function , 2017 .

[12]  Ramin Gholizadeh,et al.  Fuzzy E-Bayesian and Hierarchical Bayesian Estimations on the Kumaraswamy Distribution Using Censoring Data , 2016, Int. J. Fuzzy Syst. Appl..

[13]  Hassan M. Okasha,et al.  E-Bayesian estimation for the Lomax distribution based on type-II censored data , 2014 .

[14]  A. Zellner.  Bayesian Estimation and Prediction Using Asymmetric Loss Functions , 1986 .

[15]  Stephen P. Brooks,et al.  Markov chain Monte Carlo method and its application , 1998 .

[16]  Arnold Zellner,et al.  Hierarchical Bayesian analysis of the seemingly unrelated regression and simultaneous equations models using a combination of direct Monte Carlo and importance sampling techniques , 2010 .

[17]  Ming Han.  E-Bayesian estimation of the reliability derived from Binomial distribution , 2011 .

[18]  Z. F. Jaheen,et al.  E-Bayesian estimation for the Burr type XII model based on type-2 censoring , 2011 .

[19]  Ming Han,et al.  E-Bayesian estimation and hierarchical Bayesian estimation of failure rate , 2009 .

[20]  Christophe Andrieu,et al.  A tutorial on adaptive MCMC , 2008, Stat. Comput..

[21]  James O. Berger,et al.  Statistical Decision Theory and Bayesian Analysis, Second Edition , 1985 .

[22]  A. Kiapour,et al.  Bayes, E-Bayes and Robust Bayes Premium Estimation and Prediction under the Squared Log Error Loss Function , 2018, Journal of the Iranian Statistical Society.

[23]  Ming Han.  The E-Bayesian and hierarchical Bayesian estimations of Pareto distribution parameter under different loss functions , 2017 .

[24]  Sajid Ali,et al.  A study of the effect of the loss function on Bayes Estimate, posterior risk and hazard function for Lindley distribution , 2013 .

[25]  D. Lindley,et al.  Bayes Estimates for the Linear Model , 1972 .

[26]  Arnold Zellner,et al.  Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions , 1994 .

[27]  Ming Zhao,et al.  An integrated approach to estimate storage reliability with initial failures based on E-Bayesian estimates , 2017, Reliab. Eng. Syst. Saf..

[28]  A. Stein,et al.  Hierarchical Bayesian modeling of the space‐time diffusion patterns of cholera epidemic in Kumasi, Ghana , 2011 .