DEPENDENCE OF THE FIDUCIAL ARGUMENT ON THE SAMPLING RULE

Suppose that we have a series of observations, given to have been drawn independently from a common chance distribution of specified form depending on a single unknown parameter 6. Suppose the observations have been takeni according to some sampling rule such that the eventual lnumber of observations does lnot depend on 0 except possibly through the observations themselves. All recognized kinds of sequential rule, including fixedsample-size rules, satisfy this condition. (A type of sanmpling rule to be excluded would be one where the number of observations depended on other observations subsequently suppressed.) It is well kniown that the likelihood function of 0, given the observations, is independent of the sampling rule. For in any expression for the chance of observing what has actually been observed, the sampling rule only enters as a factor independent of 6, which is therefore irrelevant when the chances corresponding to different values of 6 are compared. It follows that the posterior probability distribution for 0 derived by Bayes's theorem from some given prior distribution is also independent of the sampling rule. (I shall refer to this use of Bayes's theorem as 'the Bayesian argument', for short.) On the other hand, in order to make a significance test (for instance, a test of whether the specified form of parent distribution is reasonably compatible with the observations), we need to take the sampling rule into account, in general. The same is true of other common statistical procedures; for examples and references, see Anscombe (1954). The purpose of this note is to show that in order to apply Fisher's fiducial argument (as recently re-expounded by him, 1956) we need to consider the sampling rule, just as for significance tests. This is interesting for two reasons. First, Fisher has stressed that a fiducial distribution is to be interpreted in exactly the same way as if it had been derived by a Bayesian argument, and indeed a fiducial distribution may be taken as prior distribution for incorporation in a subsequent Bayesian argument. Moreover, when the fiducial argument is for mathematical reasons not available, Fisher suggests that the likelihood function may be considered directly instead. Thus, what the fiducial argument is likened to and intimately associated with, and what is suggested as a substitute for it when it is not available, both differ from it in regard to dependence on the sampling rule. Secondly, Lindley (1957) has shown that, if the common chance distribution of the observations referred to in the opening sentence above is a member of the exponential family, then the fiducial argument does not have a property of consistency evidently desired for it by Fisher unless the resulting distribution for 6 is actually a posterior Bayes distribution. The examples below show that, in so far as fiducial distributions can be identified with posterior Bayes distributions, the corresponding prior distributions depend on the