Reference Priors for the General Location-Scale Model

The reference prior algorithm (Berger and Bernardo 1992) is applied to multivariate location-scale models with any regular sampling density, where we establish the irrelevance of the usual assumption of Normal sampling if our interest is in either the location or the scale. This result immediately extends to the linear regression model. 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. However, if our parameter of interest is not a one-to-one transformation of either location or scale, the choice of the sampling density is generally shown to intervene.