9 - Confidence distributions in higher dimensions

Summary

This chapter is concerned with confidence distributions for vector parameters, defined over nested families of regions in parameter space. For a given degree of confidence the region with that confidence is a confidence region in the sense of Neyman. Such regions are often called simultaneous confidence regions. They form level sets for the confidence curve representing the confidence distribution. Analytic confidence curves for vector parameters, and also for parameters of infinite dimensions, for example, all linear functions of the mean parameter vector in the normal distribution, are available in some cases. Some of these are reviewed and certain generalisations are discussed.

Introduction

Fisher's general concept of fiducial distribution is difficult in higher dimensions. Joint fiducial distributions are not subject to ordinary probability calculus. Marginals and other derived distributions need in fact not be fiducial; cf. also Chapter 6. For vector parameters one must therefore settle for a less ambitious construct to capture the inferential uncertainty. Pitman (1939, 1957) noted that the fiducial probabilities in higher dimensions must be restricted to sets of specific forms, to avoid inconsistencies. One consequence of this insight is that only certain types of dimension reduction by integrating the higher dimensional fiducial density are valid. See Chapter 6 for examples of inconsistencies when not observing these restrictions, which usually are hard to specify.

Neyman (1941) was more restrictive. He looked only for a confidence region of specific degree. We lean towards Neyman and define confidence distributions in higher dimensions as confidence distributed over specified nested families of regions.

We first look at confidence distributions for the mean vector μ in a multinormal distribution obtained from a sample, of dimension p, say. When the covariance matrix is known, the confidence distribution is simply, as will be seen, a multivariate normal distribution about the sample mean. This is obtained from a natural multivariate pivot, yielding the cumulative distribution function C(μ) assigning confidence to intervals {m: m ≤ μ﹜ in R^p. Thus confidence is also assigned to intervals and rectangles in R^p. The fiducial debate clarified that fiducial probability and hence confidence cannot be extended to the Borel sets. The confidence density for μ is thus not particularly useful, although directly obtained. Though integration over intervals yields valid confidence statements, integration over other sets might not do so.