3 - Confidence distributions

Summary

This chapter develops the notion of confidence distributions for one-dimensional focus parameters. In this one-dimensional case confidence distributions are Fisher's fiducial distributions. Confidence intervals are spanned by quantiles of the confidence distribution and one-sided p-values are cumulative confidences. Thus confidence distributions are a unifying form for representing frequentist inference for the parameter in question. Relevant graphical representations include the cumulative confidence distribution as well as its density, and the confidence curve, each providing a statistical summary of the data analysis directly pertaining to the focus parameter. We develop and examine basic methods for extracting exact or approximate confidence distributions from exact or approximate pivots, as well as from likelihood functions, and provide examples and illustrations. The apparatus is also developed to work for discrete parameters.

Introduction

The framework for the present chapter is the familiar one for parametric models, pertaining to observations Y stemming from a statistical model P_θ with θ belonging to some p-dimensional parameter region Ω. In Chapter 2 we summarised various basic classical methodologies for such models, emphasising methods based on the likelihood function. We also paid particular attention to the important type of situation is which there is a natural and typically context-driven focus parameter ψ of interest, say ψ =a(θ), and provided methods and recipes for inference related to such a ψ, including estimation, assessment of precision, testing and confidence intervals.

The purpose of the present chapter is to introduce one more general-purpose form of inference, that of confidence distributions for focus parameters under study. As such the chapter relates to core material in the non-Bayesian tradition of parametric statistical inference stemming from R. A. Fisher and J. Neyman. Confidence distributions are the Neymanian interpretation of Fisher's fiducial distributions; see also the general discussion in Chapter 1.

Confidence distributions are found from exact or approximate pivots and from likelihood functions. Pivots are functions piv(Y, ψ) of the data and focus parameter with a distribution that is exactly or approximately independent of the parameter; cf. Definition 2.3. Pivots are often related to likelihood functions, and a confidence distribution might then be obtained either from the pivot or from the likelihood.