A Fiducial Argument with Application to Survey Sampling

Summary Sprott (1960) proposed certain restrictions on distributions a posteriori in terms of the likelihood function. Hacking (1965) put forward a criterion for irrelevance restricting the choice of the pivots in terms of the likelihoods. This paper presents an extended application of Hacking's criterion to accommodate ancillary statistics. This extension is utilized to obtain the fiducial distribution of the mean of a survey (finite) population based on the given random sample. The fiducial distribution so obtained is inseparably tied to the method of randomization and as such it provides an interpretation of “randomization”. In contrast, as shown by the author (1966a, b, c), the Bayes posterior of the population mean does not depend on “randomization”. This is further discussed in Section 8. An alternative method of obtaining the fiducial distribution via Bayes's theorem is explored in Section 9.