  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 fiducial distribution is well known, we propose to
use the geometric mean of the "extreme cases" 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.
