On robust Bayesian analysis for location and scale parameters

Dawid (1973, Biometrika 60, 664-666) stated conditions in the univariate lo cation model with known scale parameter needed for there to be either vanis hing likelihood or prior influence on the posterior distribution when there is a conflict between likelihood and prior. More recently, Pericchi and Sa nso (1995, Biometrika 82, 223-225) noted that there are distributions that partially satisfy Dawid's conditions but have bounded rather than vanishing influence on the posterior distribution. In this paper, we present the ext ension of these results for the location and scale model using the multivar iate upsilon-spherical distributions. We show that when the upsilon(.) = \\ .\\ function is a norm, the \\ \\-spherical distributions, exponential powe r, and logistic power provide a robust analysis for the location model with known scale parameter, whereas Student's power t provides a robust analysi s for the location and scale model. Robust analyses are illustrated for nor mal-gamma prior location and scale models. Numerical computations are imple mented via the Gibbs sampler. (C) 1999 Academic Press.