A Neyman-Pearson-Wald View of Fiducial Probability

Abstract

This is an examination of Fisher’s early papers on Fiducial probability, from the performance point of view. A Fiducial distribution for a real- valued parameter θ is identified as a distribution-function-valued statistic (each observed value is a probability distribution on the parameter space) such that for each γ between 0 and 1 the γ-quantile of this distribution has probability γ of exceeding θ. The view taken is that there are usually many possible Fiducial distributions, and that the problem is to find one that is in some sense optimal. It is shown that Fiducial distributions can be found for discrete random variables using randomization: this is done for the Binomial (n,p) by inverting the uniformly most powerful one-sided tests. The resulting Fiducial distribution has the corresponding optimum property that its γ-quantile, subject to having probability γ of exceeding the truep, has uniformly minimum probability of exceeding each p ₁ > p.