A geometric formulation of fiducial probability

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.