Convolution oft-densities with application to Bayesian inference for a normal mean and scientific reporting

Tsai-Hung Fan, Purdue University

Abstract

An exact formula of the convolution of two t densities with odd degrees of freedom is derived. From the viewpoint of Bayesian robustness in the basic normal inference problem, a Cauchy prior is of special interest. For this case, in addition to the exact formulas for the even sample size, interpolation formulas for the marginal, posterior mean and posterior variance are derived when the sample size is odd. In comparison with the IMSL numerical integration subroutine, our algorithms turn out to be substantially more efficient and accurate. Theoretical behavior of the posterior density, the posterior mean, and the posterior variance is studied when the parameters of the t prior are chosen to be extreme. Of particular interest is the study of bimodality of the posterior. A sufficient condition for the posterior density being unimodal is also given. Finally, useful approximate formulas for the marginal, posterior c.d.f. and posterior mean are presented. Statistical applications considered include testing a point null hypothesis, one sided testing, estimation, and credible sets. A convenient way of presenting information for the statistical consumer is to give contour graphs of the Bayes factor, posterior mean, variance, etc., with respect to the prior parameters.

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics

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