Probability matching priors for the bivariate normal distribution.

摘要

In practice, most Bayesian analyses are performed with so called "non-informative" priors. This is especially so when there is little or no prior information, and yet the Bayesian technique can lead to solutions satisfactory from both the Bayesian and the frequentist perspectives. The study of probability matching priors ensuring, upto the desired order of asymptotics, the approximate frequentist validity of posterior credible sets has received significant attention in recent years. In this dissertation we develop some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, the ratio of one of the conditional variances to the marginal variance of the other variable, the correlation coefficient and the ratio of the standard deviations. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. Various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior density matching, and matching via inversion of test statistics are used.;One particular prior is found which meets all the matching criteria individually for the regression coefficient, the generalized variance and the ratio of one of the conditional variances to the marginal variance of the other variable. For the correlation coefficient though, each matching criterion leads to a different prior. There however, does not exist a prior that satisfies the matching via distribution functions criterion in this case. Finally, a general class of priors have been obtained for inference about the ratio of standard deviations.;The propriety of the resultant posteriors is proved in each case under mild conditions and simulation results suggest that the approximations are valid even for moderate sample sizes. Further, several likelihood based methods have been considered for the correlation coefficient. One common feature of all these modified likelihoods is that they are all dependent on the data only through the sample correlation coefficient r. (Full text of this dissertation may be available via the University of Florida Libraries web site. Please check http://www.uflib.ufl.edu/etd.html).