Confidence distribution

In statistical inference, the concept of a confidence distribution (CD) has often been loosely referred to as a distribution function on the parameter space that can represent confidence intervals of all levels for a parameter of interest.

Historically, it has typically been constructed by inverting the upper limits of lower sided confidence intervals of all levels, and it was also commonly associated with a fiducial[1] interpretation (fiducial distribution), although it is a purely frequentist concept.

[3] In recent years, there has been a surge of renewed interest in confidence distributions.

In particular, recent research suggests that it encompasses and unifies a wide range of examples, from regular parametric cases (including most examples of the classical development of Fisher's fiducial distribution) to bootstrap distributions, p-value functions,[5] normalized likelihood functions and, in some cases, Bayesian priors and Bayesian posteriors.

[6] Just as a Bayesian posterior distribution contains a wealth of information for any type of Bayesian inference, a confidence distribution contains a wealth of information for constructing almost all types of frequentist inferences, including point estimates, confidence intervals, critical values, statistical power and p-values,[7] among others.

Some recent developments have highlighted the promising potentials of the CD concept, as an effective inferential tool.

According to Fraser,[9] the seed (idea) of confidence distribution can even be traced back to Bayes (1763)[10] and Fisher (1930).

[13] It is also believed that these "unproductive disputes" and Fisher's "stubborn insistence"[13] might be the reason that the concept of confidence distribution has been long misconstrued as a fiducial concept and not been fully developed under the frequentist framework.

[15][16][page needed] In particular, Efron stated that this distribution "assigns probability 0.05 to θ lying between the upper endpoints of the 0.90 and 0.95 confidence interval, etc."

[16] In the classical literature,[3] the confidence distribution function is interpreted as a distribution function of the parameter θ, which is impossible unless fiducial reasoning is involved since, in a frequentist setting, the parameters are fixed and nonrandom.

To interpret the CD function entirely from a frequentist viewpoint and not interpret it as a distribution function of a (fixed/nonrandom) parameter is one of the major departures of recent development relative to the classical approach.

The nice thing about treating confidence distributions as a purely frequentist concept (similar to a point estimator) is that it is now free from those restrictive, if not controversial, constraints set forth by Fisher on fiducial distributions.

[6][14] The following definition applies;[12][17][18] Θ is the parameter space of the unknown parameter of interest θ, and χ is the sample space corresponding to data Xn={X1, ..., Xn}: Also, the function H is an asymptotic CD (aCD), if the U[0, 1] requirement is true only asymptotically and the continuity requirement on Hn(•) is dropped.

In nontechnical terms, a confidence distribution is a function of both the parameter and the random sample, with two requirements.

The second requirement (R2) sets a restriction on the function so that inferences (point estimators, confidence intervals and hypothesis testing, etc.)

This is similar to the restrictions in point estimation to ensure certain desired properties, such as unbiasedness, consistency, efficiency, etc.

[18] Unlike the classical fiducial inference, more than one confidence distributions may be available to estimate a parameter under any specific setting.

Also, unlike the classical fiducial inference, optimality is not a part of requirement.

Depending on the setting and the criterion used, sometimes there is a unique "best" (in terms of optimality) confidence distribution.

given by satisfy the two requirements in the CD definition, and they are confidence distribution functions for μ.

is optimal in terms of producing the shortest confidence intervals at any given level.

Let ρ denotes the correlation coefficient of a bivariate normal population.

This is also the posterior density of a Bayes matching prior for the five parameters in the binormal distribution.

[21] The confidence distribution coincides in this case with the Bayesian posterior using the right Haar prior.

in an infinite-dimensional Hilbert space, but in this case the confidence distribution is not a Bayesian posterior.

Some authors have proposed using them for graphically viewing what parameter values are consistent with the data, instead of coverage or performance purposes.

[6][22] Denote by the probability mass of a set C under the confidence distribution function

This ps(C) is called "support" in the CD inference and also known as "belief" in the fiducial literature.

See Figure 1 from Xie and Singh (2011)[6] for a graphical illustration of the CD inference.

A few statistical programs have implemented the ability to construct and graph confidence distributions.