On Robust Bayesian Analysis for Location and Scale Parameters

Dawid (1973,Biometrika60, 664–666) stated conditions in the univariate location model with known scale parameter needed for there to be either vanishing likelihood or prior influence on the posterior distribution when there is a conflict between likelihood and prior. More recently, Pericchi and Sansó (1995,Biometrika82, 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 extension of these results for the location and scale model using the multivariatev-spherical distributions. We show that when thev(·)=‖·‖ function is a norm, the ‖‖-spherical distributions, exponential power, and logistic power provide a robust analysis for the location model with known scale parameter, whereas Student's powertprovides a robust analysis for the location and scale model. Robust analyses are illustrated for normal-gamma prior location and scale models. Numerical computations are implemented via the Gibbs sampler.