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, a 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, an analytical method or the Gibbs sampler is used to construct the joint fiducial distribution of all the parameters of the model concerned on the basis of first determining the full conditional fiducial distributions for these parameters. Although the resulting theory is classified as being "subjective", it is maintained that this is simply due to the argument that all probability statements made about fixed but unknown parameters must be inherently subjective. In particular, a systematic framework is used to reason 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.