Advances and Applications in Statistics
Volume 14, Issue 2, Pages 117 - 143
(February 2010)
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BAYESIAN ANALYSIS FOR THE WEIBULL PARAMETERS BY USING NONINFORMATIVE PRIOR DISTRIBUTIONS
Fernando A. Moala
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Abstract: Because there is not any precise definition of the concept of noninformative prior, there are in the Bayesian literature several forms of formulating noninformative priors, for instance, Jeffreys [8], MDIP (Zellner [17, 18, 19]), Tibshirani [15], reference (Bernardo [3]) and many others. A study to check if these priors lead to the same posterior inference for small samples is of great practical interest. Thus, in this paper noninformative priors are derived for the scale and shape parameters a and b in the Weibull distribution and a fully Bayesian analysis is developed. These priors are compared by examining the frequentist coverage probabilities of Bayesian posterior intervals and their lengths. Two-sided credible intervals for a and b are then constructed for the noninformative priors and their frequentist coverage can be evaluated by Markov Chain Monte Carlo (MCMC) algorithm. A simulation study with moderately small samples shows that the coverage probability of credible intervals for reference prior is more accurate compared to those obtained by other priors. Also, the reference prior is shown to be a Tibshirani�s prior when both parameters are of interest. Besides, it is shown that the priors considered in this study are the same when the shape parameter b is known. Finally, in this paper we also give an overview of the application of Bayesian approach in reliability investigations by using these priors. Numerical inferences are illustrated for the parameters considering a data set of size �and the comparisons are examined by using numerical integration and MCMC. |
Keywords and phrases: Bayesian, Weibull, posterior, prior, distribution, MDIP, reference prior, information. |
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