Fiducial Distributions for Real Exponential Families

The fiducial argument was introduced by Fisher in order to obtain distributions for unknown parameters without the need of a bayesian perspective. In recent years, a certain interest has grown for fiducial inference. In this paper we are using a result obtained by Petrone and Veronese in order to construct a fiducial distribution for the parameter of a discrete or continuous real exponential family in a simple and quite general manner. We identify the families for which a fiducial distribution can be seen as a posterior with respect to a (improper) prior, thus completing previous results by Lindley and we demonstrate that such a prior belongs to the conjugate family. Some further results on the fiducial distribution are discussed.