Fiducial theory and optimal inference

TL;DR

This paper demonstrates that fiducial distributions in group and quasigroup models can determine optimal frequentist inference procedures, offering an alternative to Bayesian methods without relying on invariant measures.

Contribution

It generalizes fiducial theory to a broader class of models and proposes fiducial algorithms as a viable alternative for constructing frequentist inference procedures.

Findings

Fiducial distribution determines optimal equivariant inference.

The proof does not depend on invariant measures.

Fiducial algorithms can serve as alternatives to Bayesian methods.

Abstract

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.