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.[2] A confidence distribution is NOT a probability distribution function of the parameter of interest, but may still be a function useful for making inferences.[3]
In recent years, there has been a surge of renewed interest in confidence distributions.[3] In the more recent developments, the concept of confidence distribution has emerged as a purely frequentist concept, without any fiducial interpretation or reasoning. Conceptually, a confidence distribution is no different from a point estimator or an interval estimator (confidence interval), but it uses a sample-dependent distribution function on the parameter space (instead of a point or an interval) to estimate the parameter of interest.
A simple example of a confidence distribution, that has been broadly used in statistical practice, is a bootstrap distribution.[4] The development and interpretation of a bootstrap distribution does not involve any fiducial reasoning; the same is true for the concept of a confidence distribution. But the notion of confidence distribution is much broader than that of a bootstrap distribution. 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.[3]
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