Objective Bayesian analysis for bivariate Marshall-Olkin exponential distribution

The Bayesian estimators for the unknown parameters of the bivariate Marshall-Olkin exponential distribution under noninformative priors have been considered and several reference priors have been derived. A class of priors is found by matching the coverage probability of one-side Bayesian credible intervals with the corresponding frequentist coverage probabilities. It is noted that some of the reference priors are also matching priors and the posterior distributions based on the reference priors and matching priors are proper. Closed forms of Bayesian estimators are obtained with respect to the quadratic loss function. Gibbs sampling is utilized to obtain the credible intervals and coverage probabilities of parameters. Comparisons in the efficiency of the maximum likelihood estimators and Bayesian estimators under different reference priors and matching priors for various sample sizes have been done by Monte Carlo simulations. A real data set is analyzed for illustrative purpose.

[1]  James O. Berger,et al.  Estimating a Product of Means: Bayesian Analysis with Reference Priors , 1989 .

[2]  Keying Ye,et al.  Reference Prior Bayesian Analysis for Normal Mean Products , 1995 .

[3]  J. J. Higgins,et al.  Estimation and Hypothesis Testing for the Parameters of a Bivariate Exponential Distribution , 1972 .

[4]  Kk,et al.  Handbook of Statistics 7: Quality Control and Reliability , 1998 .

[5]  Edsel A. Peña,et al.  Bayes Estimation for the Marshall-Olkin Exponential Distribution , 1990 .

[6]  E. Gumbel.  Bivariate Exponential Distributions , 1960 .

[7]  Simos G. Meintanis.  Test of fit for Marshall–Olkin distributions with applications , 2007 .

[8]  J. E. Freund.  A Bivariate Extension of the Exponential Distribution , 1961 .

[9]  S. J. Press,et al.  Bayesian parameter and reliability estimation for a bivariate exponential distribution parallel sampling , 1984 .

[10]  J. Berger.  The case for objective Bayesian analysis , 2006 .

[11]  Henry W. Block,et al.  A Continuous, Bivariate Exponential Extension , 1974 .

[12]  Asit P. Basu,et al.  23 Multivariate exponential distributions and their applications in reliability , 1988 .

[13]  Dimitris Karlis,et al.  ML estimation for multivariate shock models via an EM algorithm , 2003 .

[14]  F. Downton.  Bivariate Exponential Distributions in Reliability Theory , 1970 .

[15]  Debasis Kundu,et al.  Bivariate generalized exponential distribution , 2009, J. Multivar. Anal..

[16]  Gert Heinrich,et al.  Parameter estimation for a bivariate lifetime distribution in reliability with multivariate extensions , 1995 .

[17]  Jayanta K. Ghosh,et al.  On priors providing frequentist validity for Bayesian inference , 1995 .

[18]  N. L. Johnson,et al.  Continuous Multivariate Distributions, Volume 1: Models and Applications , 2019 .

[19]  Keying Ye,et al.  Frequentist validity of posterior quantiles for a two-parameter exponential family , 1996 .

[20]  Yincai Tang,et al.  Objective Bayesian analysis of accelerated competing failure models under Type-I censoring , 2011, Comput. Stat. Data Anal..

[21]  Barry C. Arnold.  Parameter Estimation for a Multivariate Exponential Distribution , 1968 .

[22]  David D. Hanagal,et al.  Large sample tests of independence for absolutely continuous bivariate exponential distribution , 1991 .

[23]  I. Olkin,et al.  A Multivariate Exponential Distribution , 1967 .

[24]  A. Singh.  Exponential Distribution: Theory, Methods and Applications , 1996 .

[25]  Dongchu Sun.  A note on noninformative priors for Weibull distributions , 1997 .

[26]  Creasy Problem,et al.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[27]  Malay Ghosh,et al.  Some remarks on noninformative priors , 1995 .

[28]  Frank Proschan,et al.  Estimating the Parameters of a Bivariate Exponential Distribution in Several Sampling Situations. , 1973 .

[29]  Yincai Tang,et al.  Optimal step-stress test under Type-I censoring for multivariate exponential distribution , 2012 .

[30]  G. Datta.  On priors providing frequentist validity of Bayesian inference for multiple parametric functions , 1996 .