Further Examples of Inconsistencies in the Fiducial Argument

Fisher wrote as though the fiducial argument were a well-defined form of reasoning: witness phrases like "the fiducial argument itself," (Fisher (1956), p. 120) and "a genuine fiducial argument" (Fisher (1956), p. 172). However, attempts to extend the set of examples of the fiducial argument beyond the set personally approved by Fisher have often run into the difficulty that several inconsistent fiducial arguments appeared to be available for a single situation, e.g., the pair of papers by Creasy (1954) and Fieller (1954), Mauldon (1955), Tukey (1957) and Brillinger (1962). Apparently in response to the Mauldon example, Fisher intimated that the joint fiducial distribution of several parameters should be built up "rigorously by a step by step process" (Fisher (1956), p. 172). Fisher's comment is rather confusing because the Mauldon approach does appear to be a step by step approach. Brillinger (1962) presented an artificial two-parameter example in which Fisher's type of step by step approach can be applied in inconsistent ways. In Section 2 I will show that a step by step approach leads to alternative inconsistent answers even in the basic two parameter situation of sampling from the normal distribution. In Section 3 I will demonstrate a difference between the fiducial distribution for the means $(\mu_1, \mu_2)$ of a bivariate normal distribution given by Fisher (1954) and the marginal distribution of $(\mu_1, \mu_2)$ under the joint fiducial distribution of all five parameters of the bivariate normal given by Fisher (1956).