  It is shown that the fiducial distribution in a group model, or more
generally a quasigroup model, determines the optimal equivariant frequentist
inference procedures. The proof does not rely on existence of invariant
measures, and generalizes results corresponding to the choice of the right Haar
measure as a Bayesian prior. Classical and more recent examples show that
fiducial arguments can be used to give good candidates for exact or approximate
confidence distributions. It is here suggested that the fiducial algorithm can
be considered as an alternative to the Bayesian algorithm for the construction
of good frequentist inference procedures more generally.
