A geometric formulation of fiducial probability

TL;DR

This paper introduces a geometric approach to fiducial probability, providing conditions for the existence of fiducial distributions based on the properties of the underlying random distributions, and compares it to the pivotal quantity method.

Contribution

It presents a new geometric formulation of fiducial probability and characterizes when fiducial distributions exist based on distribution properties.

Findings

Fiducial distribution exists if distributions are non-intersecting and complete.

Complete characterization of distributions leading to fiducial distributions.

Analysis of intersecting and incomplete distributions in fiducial theory.

Abstract

The geometric formulation of fiducial probability employed in this paper is an improvement over the usual pivotal quantity formulation. For a single parameter and single observation, the new formulation is based on the geometric properties of an ordinary two variable function and its surface representation. The following theorem is proved: A fiducial distribution for the continuous parameter $\theta$ exists if and only if (i) the continuous random probability distributions of $x$ for different $\theta$'s are non-intersecting, and (ii) the random distributions are complete, i.e. at the extreme values of $\theta$ the limiting probability distributions are zero and one for all $x$. The proof yields also a complete characterization of random distributions that lead to fiducial distributions. The paper also treats intersecting distributions and non-intersecting incomplete distributions. The latter, which are frequently encountered in a null hypothesis, are shown to be associated with intersecting "composite" distributions. An appendix compares the pivotal and geometric formulations.