Discussion of ‘Nonparametric generalized fiducial inference for survival functions under censoring’

Abstract

1. Fiducial inference We expect that many readers will not be familiar with fiducial inference. This is in contrast to the well-founded alternatives given by Bayesian and classical inference known to every statistician today. Fiducial inference has not yet been established as a general theory, but there has been considerable progress over the past few decades, as demonstrated by the paper of Cui & Hannig (2019). To discuss their contribution, we first provide a context for fiducial inference as we see it today. The original fiducial argument of Fisher (1930, p. 532) starts by considering the relation ... where F is the cumulative distribution function for the observation x ⁠ . Fisher focused on the particular case where x is the empirical correlation of a sample of size n from the bivariate Gaussian distribution. In this case F is strictly decreasing from 1 down to 0 as a function of the unknown correlation θ ⁠ . From this, Fisher argued that 1 − F ( x | θ ) is the cumulative fiducial distribution for θ ⁠ , and that π x ( θ ) = − ∂ θ F ( x | θ ) is the fiducial density of θ given x ⁠ . Fisher’s argument uses the fact that ( 1) gives a correspondence between a uniform law for u and the sampling law for x ⁠ . The argument explains, in fact, that the percentiles of the fiducial distribution give confidence intervals, and hence the fiducial distribution is a confidence distribution in this case. Even though Fisher himself later abandoned this interpretation, it should be considered a pioneering work that led to the theory of confidence intervals and hypothesis testing as used today. Fisher (1930) is, as far as we know, the first paper that calculates exact confidence intervals and explains them as such.