Confidence distribution

Not to be confused with Confidence interval.

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]

  1. ^ Cite error: The named reference Fisher1930 was invoked but never defined (see the help page).
  2. ^ Cite error: The named reference cox1958 was invoked but never defined (see the help page).
  3. ^ a b c Cite error: The named reference Xie2013r was invoked but never defined (see the help page).
  4. ^ Cite error: The named reference Efron1998 was invoked but never defined (see the help page).
  5. ^ Cite error: The named reference Fraser1991 was invoked but never defined (see the help page).
  6. ^ Cite error: The named reference Xie2011 was invoked but never defined (see the help page).
  7. ^ Fraser, D. A. S. (2019-03-29). "The p-value Function and Statistical Inference". The American Statistician. 73 (sup1): 135–147. doi:10.1080/00031305.2018.1556735. ISSN 0003-1305.