Fiducial, confidence and objective Bayesian posterior distributions for a multidimensional parameter

We propose a way to construct fiducial distributions for a multidimensional parameter using a step-by-step conditional procedure related to the inferential importance of the components of the parameter. For discrete models, in which the non-uniqueness of the FD is well known, we propose to use the geometric mean of the \lq\lq extreme cases\rq \rq{} and show its good behavior with respect to the more traditional arithmetic mean. Connections with the generalized fiducial inference approach developed by Hannig and with confidence distributions are also analyzed. The suggested procedure strongly simplifies when the statistical model belongs to a subclass of the natural exponential family, called conditionally reducible, which includes the multinomial and the negative-multinomial models. Furthermore, because fiducial inference and objective Bayesian analysis are both attempts to derive distributions for an unknown parameter without any prior information, it is natural to discuss their relationships. In particular, the reference posteriors, which also depend on the importance ordering of the parameters are the natural terms of comparison. We show that fiducial and reference posterior distributions coincide in the location-scale models, and we characterize the conditionally reducible natural exponential families for which this happens. The discussion of some classical examples closes the paper.