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. It was also commonly associated with a fiducial<ref name = "Fisher1930"/> interpretation (fiducial distribution), although it is a purely frequentist concept.<ref name="cox1958" /> A confidence distribution is {{em|not}} a probability distribution function of the parameter of interest, but may still be a function useful for making inferences.<ref name="Xie2013r" />

In recent years, there has been a surge of renewed interest in confidence distributions.<ref name="Xie2013r" /> 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.<ref name="Efron1998" /> 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,<ref name = "Fraser1991"/> normalized likelihood functions and, in some cases, Bayesian priors and Bayesian posteriors.<ref name="Xie2011" />

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,<ref>{{Cite journal|last=Fraser|first=D. A. S.|date=2019-03-29|title=The p-value Function and Statistical Inference|journal=The American Statistician|volume=73|issue=sup1|pages=135–147|doi=10.1080/00031305.2018.1556735|issn=0003-1305|doi-access=free}}</ref> among others. Some recent developments have highlighted the promising potentials of the CD concept, as an effective inferential tool.<ref name="Xie2013r" />

== History ==

Neyman (1937)<ref name="Neyman1937" /> introduced the idea of "confidence" in his seminal paper on confidence intervals which clarified the frequentist repetition property. According to Fraser,<ref name = "Fraser2011"/> the seed (idea) of confidence distribution can even be traced back to Bayes (1763)<ref name="Bayes1973" /> and Fisher (1930).<ref name="Fisher1930" /> Although the phrase seems to first be used in Cox (1958).<ref>{{Cite journal|last=Cox|first=D. R.|date=June 1958|title=Some Problems Connected with Statistical Inference|url=http://projecteuclid.org/euclid.aoms/1177706618|journal=The Annals of Mathematical Statistics|language=en|volume=29|issue=2|pages=357–372|doi=10.1214/aoms/1177706618|issn=0003-4851|doi-access=free|url-access=subscription}}</ref> Some researchers view the confidence distribution as "the Neymanian interpretation of Fisher's fiducial distributions",<ref name="Schweder2002" /> which was "furiously disputed by Fisher".<ref name="Zabell1992" /> It is also believed that these "unproductive disputes" and Fisher's "stubborn insistence"<ref name="Zabell1992" /> 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.<ref name="Xie2011" /><ref name="Singh2011" /> Indeed, the confidence distribution is a purely frequentist concept with a purely frequentist interpretation, although it also has ties to Bayesian and fiducial inference concepts.

== Definition ==

=== Classical definition ===

Classically, a confidence distribution is defined by inverting the upper limits of a series of lower-sided confidence intervals.<ref name = "Cox2006a"/><ref name = "Efron1993"/>{{page needed|date=May 2011}} In particular,

{{block indent | em = 1.5 | text = For every ''α'' in (0, 1), let (−∞, ''ξ''''n''(''α'')] be a 100α% lower-side confidence interval for ''θ'', where ''ξ''''n''(''α'') = ''ξ''''n''(''X''n,α) is continuous and increasing in α for each sample ''X''''n''. Then, ''H''''n''(•) = ''ξ''''n''−1(•) is a confidence distribution for ''θ''.}}

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''." and "it has powerful intuitive appeal".<ref name="Efron1993" /> In the classical literature,<ref name="Xie2013r" /> 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.<ref name = "Xie2011"/><ref name = "Singh2011" />

=== The modern definition ===

The following definition applies;<ref name="Schweder2002" /><ref name = "Singh2001"/><ref name="Singh2005" /> ''Θ'' is the parameter space of the unknown parameter of interest ''θ'', and ''χ'' is the sample space corresponding to data '''''X'''''''n''={''X''1, ..., ''X''''n''}:

{{block indent | em = 1.5 | text = A function ''H''''n''(•) = ''H''''n''('''''X'''''''n'', •) on ''χ'' × ''Θ'' → [0, 1] is called a confidence distribution (CD) for a parameter ''θ'', if it follows two requirements: *(R1) For each given ''X''''n'' ∈ ''χ'', ''H''''n''(•) = ''H''''n''(''X''''n'', •) is a continuous cumulative distribution function on ''Θ''; *(R2) At the true parameter value ''θ'' = ''θ''0, ''H''''n''(''θ''0) ≡ ''H''''n''('''''X'''''''n'', ''θ''0), as a function of the sample '''''X'''''''n'', follows the uniform distribution ''U''[0, 1]. }} Also, the function ''H'' is an asymptotic CD ('''aCD'''), if the ''U''[0, 1] requirement is true only asymptotically and the continuity requirement on ''H''''n''(•) is dropped.

In nontechnical terms, a confidence distribution is a function of both the parameter and the random sample, with two requirements. The first requirement (R1) simply requires that a CD should be a distribution on the parameter space. The second requirement (R2) sets a restriction on the function so that inferences (point estimators, confidence intervals and hypothesis testing, etc.) based on the confidence distribution have desired frequentist properties. This is similar to the restrictions in point estimation to ensure certain desired properties, such as unbiasedness, consistency, efficiency, etc.<ref name="Xie2011" /><ref name="Xie2009" />

A confidence distribution derived by inverting the upper limits of confidence intervals (classical definition) also satisfies the requirements in the above definition and this version of the definition is consistent with the classical definition.<ref name="Singh2005" />

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. But sometimes there is no optimal confidence distribution available or, in some extreme cases, we may not even be able to find a meaningful confidence distribution. This is not different from the practice of point estimation.

=== A definition with measurable spaces ===

A confidence distribution<ref>{{Cite journal|last=Taraldsen|first=Gunnar|date=2021|title=Joint Confidence Distributions|url=http://rgdoi.net/10.13140/RG.2.2.33079.85920|language=en|doi=10.13140/RG.2.2.33079.85920}}</ref> <math>C</math> for a parameter <math>\gamma</math> in a measurable space is a distribution estimator with <math>C(A_p) = p</math> for a family of confidence regions <math>A_p</math> for <math>\gamma</math> with level <math>p</math> for all levels <math>0 . The family of confidence regions is not unique.<ref name="Liu 1–19">{{Cite journal|last1=Liu|first1=Dungang|last2=Liu|first2=Regina Y.|last3=Xie|first3=Min-ge|date=2021-04-30|title=Nonparametric Fusion Learning for Multiparameters: Synthesize Inferences From Diverse Sources Using Data Depth and Confidence Distribution|url=https://www.tandfonline.com/doi/full/10.1080/01621459.2021.1902817|journal=Journal of the American Statistical Association|volume=117 |issue=540 |language=en|pages=2086–2104|doi=10.1080/01621459.2021.1902817|s2cid=233657455 |issn=0162-1459}}</ref> If <math>A_p</math> only exists for <math>p \in I \subset (0,1)</math>, then <math>C</math> is a confidence distribution with level set <math>I</math>. Both <math>C</math> and all <math>A_p</math> are measurable functions of the data. This implies that <math>C</math> is a random measure and <math>A_p</math> is a random set. If the defining requirement <math>P(\gamma \in A_p) \ge p</math> holds with equality, then the confidence distribution is by definition exact. If, additionally, <math>\gamma</math> is a real parameter, then the measure theoretic definition coincides with the above classical definition.

==Examples==

=== Example 1: Normal mean and variance ===

Suppose a normal sample ''X''''i'' ~ ''N''(''μ'', ''σ''2), ''i'' = 1, 2, ..., ''n'' is given.

==== Known Variance ''σ''2 ====

Let, ''Φ'' be the cumulative distribution function of the standard normal distribution, and <math> F_{t_{n-1}} </math> the cumulative distribution function of the Student <math> t_{n-1} </math> distribution. Both the functions <math>H_\mathit{\Phi}(\mu)</math> and <math>H_t(\mu)</math> given by

<math display="block"> H_\Phi(\mu) = \mathit{\Phi}{\left(\frac{\sqrt{n}(\mu-\bar{X})}{\sigma}\right)} , \quad\text{and}\quad H_t(\mu) = F_{t_{n-1}}{\left(\frac{\sqrt{n}(\mu-\bar{X})}{s}\right)} , </math>

satisfy the two requirements in the CD definition, and they are confidence distribution functions for ''μ''.<ref name="Xie2013r" /> Furthermore,

<math display="block"> H_A(\mu) = \mathit{\Phi}{\left(\frac{\sqrt{n}(\mu-\bar{X})}{s}\right)}</math>

satisfies the definition of an asymptotic confidence distribution when ''n''→∞, and it is an asymptotic confidence distribution for ''μ''. The uses of <math>H_{t}(\mu)</math> and <math>H_{A}(\mu)</math> are equivalent to state that we use <math>N(\bar{X},\sigma^2)</math> and <math>N(\bar{X},s^2)</math> to estimate <math>\mu</math>, respectively.

==== Unknown variance ====

For the parameter ''μ'', since <math>H_\mathit{\Phi}(\mu)</math> involves the unknown parameter ''σ'' and it violates the two requirements in the CD definition, it is no longer a "distribution estimator" or a confidence distribution for ''μ''.<ref name="Xie2013r" /> However, <math>H_{t}(\mu)</math> is still a CD for ''μ'' and <math>H_{A}(\mu)</math> is an aCD for ''μ''.

For the parameter ''σ''2, the sample-dependent cumulative distribution function

<math display="block">H_{\chi^2}(\theta) = 1 - F_{\chi^2_{n-1}}{\left((n-1)s^2/\theta\right)}</math>

is a confidence distribution function for ''σ''2.<ref name = "Xie2011"/> Here, <math> F_{\chi^2_{n-1}} </math> is the cumulative distribution function of the <math> \chi^2_{n-1} </math> distribution .

In the case when the variance ''σ''2 is known, <math display="inline"> H_{\mathit{\Phi}}(\mu) = \mathit{\Phi} \left(\frac{\sqrt{n}}{\sigma}(\mu-\bar{X})\right) </math> is optimal in terms of producing the shortest confidence intervals at any given level. In the case when the variance ''σ''2 is unknown, <math display="inline"> H_{t}(\mu) = F_{t_{n-1}}\left(\frac{\sqrt{n}}{s}(\mu-\bar{X})\right) </math> is an optimal confidence distribution for ''μ''.

=== Example 2: Bivariate normal correlation ===

Let ''ρ'' denotes the correlation coefficient of a bivariate normal population. It is well known that Fisher's ''z'' defined by the Fisher transformation:

has the limiting distribution <math display="inline">N{\left({1 \over 2}\ln{{1+\rho}\over{1-\rho}}, {1 \over n-3}\right)}</math> with a fast rate of convergence, where ''r'' is the sample correlation and ''n'' is the sample size.

The function

<math display="block">H_n(\rho) = 1 - \mathit{\Phi}\left(\sqrt{n-3} \left({1 \over 2}\ln{1+r \over 1-r} -{1 \over 2}\ln{{1+\rho}\over{1-\rho}} \right)\right)</math>

is an asymptotic confidence distribution for ''ρ''.<ref name="Singh2007" />

An exact confidence density for ''ρ'' is<ref>{{Cite journal|last=Taraldsen|first=Gunnar|date=2021|title=The Confidence Density for Correlation|journal=Sankhya A|volume=85 |pages=600–616 |language=en|doi=10.1007/s13171-021-00267-y|s2cid=244594067 |issn=0976-8378|doi-access=free|hdl=11250/3133125|hdl-access=free}}</ref><ref>{{Cite journal|last=Taraldsen|first=Gunnar|date=2020|title=Confidence in Correlation|url=http://rgdoi.net/10.13140/RG.2.2.23673.49769|language=en|doi=10.13140/RG.2.2.23673.49769}}</ref>

<math display="block">\pi (\rho \mid r) = \frac{\nu (\nu - 1)\Gamma(\nu-1)}{\sqrt{2\pi}\Gamma(\nu + \frac{1}{2})} \left(1 - r^2\right)^{\frac{\nu - 1}{2}} \cdot \left(1 - \rho^2\right)^{\frac{\nu - 2}{2}} \cdot \left(1 - r \rho\right)^{-\nu+\frac{1}{2}} F{\left(\frac{3}{2},-\frac{1}{2}; \nu + \frac{1}{2}; \frac{1 + r \rho}{2}\right)}</math>

where <math>F</math> is the Gaussian hypergeometric function and <math>\nu = n-1 > 1</math> . This is also the posterior density of a Bayes matching prior for the five parameters in the binormal distribution.<ref>{{Cite journal|last1=Berger|first1=James O.|last2=Sun|first2=Dongchu|date=2008-04-01|title=Objective priors for the bivariate normal model|journal=The Annals of Statistics|volume=36|issue=2|doi=10.1214/07-AOS501|s2cid=14703802 |issn=0090-5364|doi-access=free|arxiv=0804.0987}}</ref>

The very last formula in the classical book by Fisher gives

<math display="block">\pi (\rho | r) = \frac{(1 - r^2)^{\frac{\nu - 1}{2}} \cdot (1 - \rho^2)^{\frac{\nu - 2}{2}}}{\pi (\nu - 2)!} \partial_{\rho r}^{\nu - 2} \left\{ \frac{\theta - \frac{1}{2}\sin 2\theta}{\sin^3 \theta} \right\}</math>

where <math> \cos \theta = -\rho r</math> and <math>0 < \theta < \pi</math>. This formula was derived by C. R. Rao.<ref>{{Cite book|last=Fisher|first=Ronald Aylmer, Sir|title=Statistical methods and scientific inference|date=1973|publisher=Hafner Press|isbn=0-02-844740-9|edition=[3d ed., rev. and enl.]|location=New York|oclc=785822}}</ref>

=== Example 3: Binormal mean ===

Let data be generated by <math>Y = \gamma + U</math> where <math>\gamma</math> is an unknown vector in the plane and <math>U</math> has a binormal and known distribution in the plane. The distribution of <math>\Gamma^y = y - U</math> defines a confidence distribution for <math>\gamma</math>. The confidence regions <math>A_p</math> can be chosen as the interior of ellipses centered at <math>\gamma</math> and axes given by the eigenvectors of the covariance matrix of <math>\Gamma^y</math>. The confidence distribution is in this case binormal with mean <math>\gamma</math>, and the confidence regions can be chosen in many other ways.<ref name="Liu 1–19"/> The confidence distribution coincides in this case with the Bayesian posterior using the right Haar prior.<ref>{{Cite journal|last1=Eaton|first1=Morris L.|last2=Sudderth|first2=William D.|date=2012|title=Invariance, model matching and probability matching|url=https://www.jstor.org/stable/42003718|journal=Sankhyā: The Indian Journal of Statistics, Series A (2008-)|volume=74|issue=2|pages=170–193|doi=10.1007/s13171-012-0018-4 |jstor=42003718 |s2cid=120705955 |issn=0976-836X|url-access=subscription}}</ref> The argument generalizes to the case of an unknown mean <math>\gamma</math> in an infinite-dimensional Hilbert space, but in this case the confidence distribution is not a Bayesian posterior.<ref name="Taraldsen">{{Cite journal|last1=Taraldsen|first1=Gunnar|last2=Lindqvist|first2=Bo Henry|date=2013-02-01|title=Fiducial theory and optimal inference|journal=The Annals of Statistics|volume=41|issue=1|doi=10.1214/13-AOS1083|s2cid=88520957 |issn=0090-5364|doi-access=free|arxiv=1301.1717}}</ref>

== Using confidence distributions for inference ==

=== Confidence interval ===

right|thumb|400px From the CD definition, it is evident that the interval <math>(-\infty, H_n^{-1}(1-\alpha)], [H_n^{-1}(\alpha), \infty)</math> and <math>[H_n^{-1}(\alpha/2), H_n^{-1}(1-\alpha/2)]</math> provide 100(1 − ''α'')%-level confidence intervals of different kinds, for ''θ'', for any ''α'' ∈ (0, 1). Also <math>[H_n^{-1}(\alpha_1), H_n^{-1}(1-\alpha_2)]</math> is a level 100(1 − ''α''1 − ''α''2)% confidence interval for the parameter ''θ'' for any ''α''1 > 0, ''α''2 > 0 and ''α''1 + ''α''2 < 1. Here, <math> H_n^{-1}(\beta) </math> is the 100''β''% quantile of <math> H_n(\theta) </math> or it solves for ''θ'' in equation <math> H_n(\theta)=\beta </math>. The same holds for a CD, where the confidence level is achieved in limit. Some authors have proposed using them for graphically viewing what parameter values are consistent with the data, instead of coverage or performance purposes.<ref>{{Cite book|last1=Cox|first1=D. R.|url=https://www.taylorfrancis.com/books/9780429170218|title=Theoretical Statistics|last2=Hinkley|first2=D. V.|date=1979-09-06|publisher=Chapman and Hall/CRC|isbn=978-0-429-17021-8|language=en|doi=10.1201/b14832}}</ref><ref>{{Cite journal|last1=Rafi|first1=Zad|last2=Greenland|first2=Sander|date=2020-09-30|title=Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise|url= |journal=BMC Medical Research Methodology|volume=20|issue=1|pages=244|doi=10.1186/s12874-020-01105-9|arxiv=1909.08579|issn=1471-2288|pmc=7528258|pmid=32998683 |doi-access=free }}</ref>

=== Point estimation ===

Point estimators can also be constructed given a confidence distribution estimator for the parameter of interest. For example, given ''H''''n''(''θ'') the CD for a parameter ''θ'', natural choices of point estimators include the median ''M''''n'' = ''H''''n''−1(1/2), the mean <math display="inline">\bar{\theta}_n = \int_{-\infty}^\infty t \, \mathrm{d}H_n(t)</math>, and the maximum point of the CD density

<math display="block">\widehat{\theta}_n=\arg\max_\theta h_n(\theta), h_n(\theta)=H'_n(\theta).</math>

Under some modest conditions, among other properties, one can prove that these point estimators are all consistent.<ref name = "Xie2011" /><ref name = "Singh2007" /> Certain confidence distributions can give optimal frequentist estimators.<ref name="Taraldsen"/>

=== Hypothesis testing ===

One can derive a p-value for a test, either one-sided or two-sided, concerning the parameter ''θ'', from its confidence distribution ''H''''n''(''θ'').<ref name="Xie2011" /><ref name="Singh2007" /> Denote by the probability mass of a set ''C'' under the confidence distribution function <math display="inline"> p_s(C) = H_n(C) = \int_C \mathrm{d} H(\theta). </math> This ''p''''s''(C) is called "support" in the CD inference and also known as "belief" in the fiducial literature.<ref name ="Kendall1974"/> We have

# For the one-sided test ''K''0: ''θ'' ∈ ''C'' vs. ''K''1: ''θ'' ∈ ''C''c, where ''C'' is of the type of (−∞, ''b''] or [''b'', ∞), one can show from the CD definition that sup''θ'' ∈ ''C''''P''''θ''(''p''''s''(''C'') ≤ ''α'') = ''α''. Thus, ''p''''s''(''C'') = ''H''''n''(''C'') is the corresponding p-value of the test. # For the singleton test ''K''0: ''θ'' = ''b'' vs. ''K''1: ''θ'' ≠ ''b'', ''P''{''K''0: ''θ'' = ''b''}(2 min{''p''''s''(''C''lo), one can show from the CD definition that ps(''C''up)} ≤ ''α'') = ''α''. Thus, 2 min{''p''''s''(''C''lo), ''p''''s''(''C''up)} = 2 min{''H''''n''(''b''), 1 − ''H''''n''(''b'')} is the corresponding p-value of the test. Here, ''C''lo = (−∞, ''b''] and ''C''up = [''b'', ∞).

See Figure 1 from Xie and Singh (2011)<ref name = "Xie2011"/> for a graphical illustration of the CD inference.

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

* R, via the <code>concurve</code>,<ref name="cran.r-project.org">{{Citation|last1=Rafi [aut|first1=Zad|title=concurve: Computes and Plots Compatibility (Confidence) Intervals, P-Values, S-Values, & Likelihood Intervals to Form Consonance, Surprisal, & Likelihood Functions|date=2020-04-20|url=https://cran.r-project.org/package=concurve|access-date=2020-05-05|last2=cre|last3=Vigotsky|first3=Andrew D.}}</ref><ref>{{Cite web|url=https://statmodeling.stat.columbia.edu/2019/05/29/concurve-plots-consonance-curves-p-value-functions-and-s-value-functions/|title=Concurve plots consonance curves, p-value functions, and S-value functions « Statistical Modeling, Causal Inference, and Social Science|website=statmodeling.stat.columbia.edu|language=en-US|access-date=2020-04-15}}</ref> <code>pvaluefunctions</code>,<ref>{{Citation|last=Infanger|first=Denis|title=pvaluefunctions: Creates and Plots P-Value Functions, S-Value Functions, Confidence Distributions and Confidence Densities|date=2019-11-29|url=https://cran.r-project.org/package=pvaluefunctions|access-date=2020-04-15}}</ref> and <code>episheet</code><ref>{{Citation|last1=Black|first1=James|title=episheet: Rothman's Episheet|date=2019-01-23|url=https://cran.r-project.org/package=episheet|access-date=2020-04-15|last2=Rothman|first2=Ken|last3=Thelwall|first3=Simon}}</ref> packages * Excel, via <code>episheet</code><ref>{{Cite web|url=http://www.krothman.org/|title=Modern Epidemiology, 2nd Edition|website=www.krothman.org|access-date=2020-04-15|archive-date=2020-01-29|archive-url=https://web.archive.org/web/20200129153412/http://www.krothman.org/|url-status=dead}}</ref> * Stata, via <code>concurve</code><ref name="cran.r-project.org" />

==See also== * Coverage probability

==References== <references>

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Category:Estimation theory Category:Parametric statistics