Some Examples with Fiducial Relevance

It has been believed by some (17, p. 204, and perhaps, by implication 21, p. 2, near line 10) that--and R. A. Fisher e.g. at the Lake Junaluska conference in 1946 has urged the desirability of determining whether--the distribution induced by a pivotal, sufficient and smoothly invertible set of quantities is unique (that is to say, the induced distribution is independent of the choice of a particular set of pivotal quantities among those sets with these properties). If true, such uniqueness would be important in connection with the theory of fiducial probability. It is the purpose of this paper to present certain examples of particular interest showing that these conditions do not provide uniqueness. The first example applies to any family of two-dimensional normal distributions with fixed and known variances and covariances. A one-parameter family of pivotal pairs of quantities are provided, such that no two of the induced distributions are the same. Each pair is sufficient, and consists of two independent quantities, each distributed according to a unit normal distribution. Each pair is shown to be smoothly invertible of every finite order. This example can be extended to the Behrens-Fisher situation. The second example is due to L. J. Savage, and exhibits a two-parameter situation where the two alternative pairs of pivotal quantities constructed according to the prescription of Segal 24 give rise to different distributions. Mauldon 19 has recently published a quite different example of nonuniqueness which is also based on the bivariate normal distribution. In his example, the means are known and the second moments are to be estimated, so that there are 3 essential parameters. The paper concludes with a reasonably complete bibliography of papers on fiducial probability.