Oct 14

3:30 pm

## Reference Priors in non-Normal Location Problems

### Carmen Fernandez & Mark Steel

Seminar

Tilburg University

The reference prior algorithm [Berger and Bernardo (1992)] is applied to location-scale models with any regular sampling density, where we establish the irrelevance of the usual assumption of Normal sampling. On the other hand, an essentially arbitrary step in the reference prior algorithm, namely the choice of the nested sequence of sets in the parameter space is seen to play a role. Our results lend an additional motivation to the often used prior proportional to the inverse of the scale parameter, as it is found to be both the independence Jeffreys' prior and the reference prior under variation independence in the sequence of sets, for any choice of the sampling density. A number of two-sample problems is analyzed in this general context, extending the Behrens-Fisher, Fieller-Creasy and product of Normal means problems outside Normality, while explicitly considering possibly differenct sizes for each sample. In these problems the choice of sampling densities can influence the form of the reference prior. Since the reference prior turns out to be improper in all cases, we examine existence of the resulting posterior distribution and its moments under sampling from scale mixtures of Normals. In the context of an empirical example, it is shown that a reference posterior analysis is numerically feasible and can display some sensitivity to the actual sampling distributions. This illustrates the practical importance of questioning the Normality assumption.

**Key Words**

Behrens-Fisher problem; Fieller-Creasy problem; Jeffreys' prior; Location-scale model; Two-sample problems.