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
interpretation (
fiducial distribution), although it is a purely frequentist concept.
A confidence distribution is NOT a probability distribution function of the parameter of interest, but may still be a function useful for making inferences.
In recent years, there has been a surge of renewed interest in confidence distributions.
In the more recent developments, the concept of confidence distribution has emerged as a purely
frequentist
Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or pro ...
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.
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,
normalized
likelihood function
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...
s and, in some cases, Bayesian
priors and Bayesian
posteriors.
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 estimate
In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown popu ...
s,
confidence intervals, critical values,
statistical power and p-values, among others. Some recent developments have highlighted the promising potentials of the CD concept, as an effective inferential tool.
History
Neyman (1937)
introduced the idea of "confidence" in his seminal paper on confidence intervals which clarified the frequentist repetition property. According to Fraser,
the seed (idea) of confidence distribution can even be traced back to Bayes (1763)
and Fisher (1930).
Although the phrase seems to first be used in Cox (1958). Some researchers view the confidence distribution as "the Neymanian interpretation of Fisher's fiducial distributions",
which was "furiously disputed by Fisher".
It is also believed that these "unproductive disputes" and Fisher's "stubborn insistence"
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.
Indeed, the confidence distribution is a purely frequentist concept with a purely frequentist interpretation, and it also has ties to Bayesian inference concepts and the fiducial arguments.
Definition
Classical definition
Classically, a confidence distribution is defined by inverting the upper limits of a series of lower-sided confidence intervals.
In particular,
: 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".
In the classical literature,
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.
The modern definition
The following definition applies;
''Θ'' is the parameter space of the unknown parameter of interest ''θ'', and ''χ'' is the sample space corresponding to data ''X''
''n''=:
: A function ''H''
''n''(•) = ''H''
''n''(''X''
''n'', •) on ''χ'' × ''Θ'' →
, 1is 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''
, 1
Also, the function ''H'' is an asymptotic CD (aCD), if the ''U''
, 1requirement 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.
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.
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
for a
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
in a
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
is a distribution
estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...
with
for a family of
confidence region In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an ''n''-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, al ...
s
for
with level
for all levels
. The family of confidence regions is not unique.
If
only exists for
, then
is a confidence distribution with level set
. Both
and all
are measurable functions of the data. This implies that
is a
random measure In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. ...
and
is a
random set. If the defining requirement
holds with equality, then the confidence distribution is by definition exact. If, additionally,
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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
sample ''X''
''i'' ~ ''N''(''μ'', ''σ''
2), ''i'' = 1, 2, ..., ''n'' is given.
(1) Variance ''σ''
2 is known
Let, ''Φ'' be the cumulative distribution function of the standard normal distribution, and
the cumulative distribution function of the Student
distribution. Both the functions
and
given by
:
satisfy the two requirements in the CD definition, and they are confidence distribution functions for ''μ''.
Furthermore,
:
satisfies the definition of an asymptotic confidence distribution when ''n''→∞, and it is an asymptotic confidence distribution for ''μ''. The uses of
and
are equivalent to state that we use
and
to estimate
, respectively.
(2) Variance ''σ''
2 is unknown
For the parameter ''μ'', since
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''.
However,
is still a CD for ''μ'' and
is an aCD for ''μ''.
For the parameter ''σ''
2, the sample-dependent cumulative distribution function
:
is a confidence distribution function for ''σ''
2.
Here,
is the cumulative distribution function of the
distribution .
In the case when the variance ''σ''
2 is known,
is optimal in terms of producing the shortest confidence intervals at any given level. In the case when the variance ''σ''
2 is unknown,
is an optimal confidence distribution for ''μ''.
Example 2: Bivariate normal correlation
Let ''ρ'' denotes the
correlation coefficient
A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
of a
bivariate normal
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One ...
population. It is well known that Fisher's ''z'' defined by the
Fisher transformation
In statistics, the Fisher transformation (or Fisher ''z''-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh).
When the sample correlation coefficient ''r'' is near 1 or -1, its distribution is high ...
:
:
has the
limiting distribution with a fast rate of convergence, where ''r'' is the sample correlation and ''n'' is the sample size.
The function
:
is an asymptotic confidence distribution for ''ρ''.
An exact confidence density for ''ρ'' is
where
is the Gaussian hypergeometric function and
. This is also the posterior density of a Bayes matching prior for the five parameters in the binormal distribution.
The very last formula in the classical book by
Fisher
Fisher is an archaic term for a fisherman, revived as gender-neutral.
Fisher, Fishers or The Fisher may also refer to:
Places
Australia
*Division of Fisher, an electoral district in the Australian House of Representatives, in Queensland
*Elect ...
gives
where
and
. This formula was derived by
C. R. Rao
Calyampudi Radhakrishna Rao FRS (born 10 September 1920), commonly known as C. R. Rao, is an Indian-American mathematician and statistician. He is currently professor emeritus at Pennsylvania State University and Research Professor at the Un ...
.
Example 3: Binormal mean
Let data be generated by
where
is an unknown vector in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
and
has a
binormal and known distribution in the plane. The distribution of
defines a confidence distribution for
. The confidence regions
can be chosen as the interior of
ellipses centered at
and axes given by the eigenvectors of the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
matrix of
. The confidence distribution is in this case binormal with mean
, and the confidence regions can be chosen in many other ways.
The confidence distribution coincides in this case with the Bayesian posterior using the right Haar prior. The argument generalizes to the case of an unknown mean
in an infinite-dimensional
Hilbert space, but in this case the confidence distribution is not a Bayesian posterior.
Using confidence distributions for inference
Confidence interval
From the CD definition, it is evident that the interval