Objective Bayesian Analysis For a Two-parameters Exponential Distribution

Document Type : Original Article

Authors
Ali Soori 1
Parviz Nasiri 2
Mehdi Jabbari Nooghabi 3
Farshin Hormozinejad 1
Mohammadreza Ghalani 1
1 Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran
2 Department of Statistics, University of Payam Noor, 19395-4697 Tehran, Iran
3 Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran.
10.22034/maco.3.1.7
Abstract
In any Bayesian inference problem, the posterior distribution is a product of the likelihood and the prior: thus, it is affected by both in cases where one possesses little or no information about the target parameters in advance. In the case of an objective Bayesian analysis, the resulting posterior should be expected to be universally agreed upon by everyone, whereas . subjective Bayesianism would argue that probability corresponds to the degree of personal belief. In this paper, we consider Bayesian estimation of two-parameter exponential distribution using the Bayes approach needs a prior distribution for parameters. However, it is difficult to use the joint prior distributions. Sometimes, by using linear transformation of reliability function of two-parameter exponential distribution in order to
get simple linear regression model to estimation of parameters. Here, we propose to make Bayesian inferences for the parameters using non-informative priors, namely the (dependent and independent) Jeffreys’ prior and the reference prior. The Bayesian estimation was assessed using the Monte Carlo method. The criteria mean square error was determined evaluate the possible impact of prior specification on estimation. Finally, an application on
a real dataset illustrated the developed procedures.

Keywords

[1] S. P. Ahmad and B. A. Bhat, Posterior Estimates of Two-Parameter Exponential Distribution using SPLUS Software, Journal of Reliability and Statistical Studies 3 (2):27-34, 2010.
[2] B. G. AL-Ani, R. S. AL-Rassam and S. N. Rashed, Bayesian Estimation for Two-Parameter Exponential Distribution Using Linear Transformation of Reliability Function, Periodicals of Engineering and Natural Sciences 8 (1):242-247, 2020.
[3] H. S. AL-Kutubi and N. A. Ibrahim, Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information, Malaysian Journal of Mathematical Sciences 3 (2):297-313, 2009.
[4] S. A. Anfinogentov, V. M. Nakariakov, D. J. Pascoe and C. R. Goddard, Solar Bayesian analysis toolkit—a new Markov chain Monte Carlo IDL code for Bayesian parameter inference, The Astrophysical Journal Supplement Series 252 (1):11, (2021).
[5] P. H. Ferreira, E. Ramos, P. L. Ramos, J. F. B. Gonzales, V. L. D. Tomazella, R. S. Ehlers, E. B. Silva and F. Louzada, Objective bayesian analysis for the lomax distribution, Statistics and Probability Letters 159:108677, 2020.
[6] C. B. Guure, N. A. Ibrahim and A. M. Ahmed, Bayesian estimation of two-parameter Weibull distribution using extension of Jeffery prior information with three loss functions, Mathematical Problems in Engineering 2012:589640, 2012.
[7] S. Kourouklis, Estimation in the 2-parameter exponential distribution with prior information, IEEE Transactions on Reliability 43 (3):446-450, 1994.
[8] K. Lam, B. K. Sinha and Z. Wu, Estimation of Parameters in a Two-Parameter Exponential Distribution using Ranked Set Sample, Ann. Inst. Statist. 46 (4):723-736, 1994.
[9] D. J. Lunn, A. Thomas, N. Best and D. Spiegelhalter, Win BUGS-a Bayesian modelling framework: concepts, structure, and extensibility, Statistics and computing 10 (4):325-337, 2000.
[10] M. Plummer, JAGS, A program for analysis of Bayesian graphical models using Gibbs sampling, Proceedings of 3rd International Workshop on Distributed Statistical Computing (DSC 2003), March 20-22, Vienna, Austria, 2003.
[11] R. Singh, S. K. Singh, U. Singh and G. P. Singh, Bayes Estimator of Generalized-Exponential parameters under LINEX Loss Function Using Lindley’s Approximation, Data Science Journal 7 (5):65-75, 2008.
[12] R. Singh, S. K. Upadhyay and U. Singh, Bayes Estimators for Two-Parameter Exponential Distribution, Common. Statist. Theory. Meth. 24 (1):227-240, 1995.
[13] R. Tanabe, and E. Hamada, Priors for the zero-modified model, Statist. Probab. Lett. 112:92–97, 2016.
[14] H. Wang, and D. Sun, Objective Bayesian analysis for a truncated model, Statistics and Probability Letters 82 (12):2125-2135, (2012).
[15] S. F. Wu, Bayesian interval estimation for the two-parameter exponential distribution based on the right type II censored sample, Symmetry 14 (2):352, 2022.
Mathematical Analysis and Convex Optimization
Volume 3, Issue 1
June 2022
Pages 61-71