Multivariate subjective fiducial inference

The aim of this paper is to firmly establish subjective fiducial inference as a rival to the more conventional schools of statistical inference, and to show that Fisher's intuition concerning the importance of the fiducial argument was correct. In this regard, methodology outlined in an earlier paper is modified, enhanced and extended to deal with general inferential problems in which various parameters are unknown. As part of this, the joint fiducial distribution of all the parameters of a given model is determined on the basis of the full conditional fiducial distributions of these parameters by using an analytical approach or a Gibbs sampling method, the latter of which does not require these conditional distributions to be compatible. Although the resulting theory is classified as being "subjective", the reason for this is essentially attributed to the argument that all probability statements made about fixed but unknown parameters must be inherently subjective. In particular, it is systematically argued that, in general, there is no need to place a great emphasis on the difference between the fiducial probabilities that can be derived using this theory and objective probabilities. Some important examples of the application of this theory are presented.