^{1}

^{*}

^{2}

^{*}

^{3}

^{1}

The reconstruction of a parameter by the measurement of a random variable depending on the parameter is one of the main tasks in statistics. In statistical inference, the concept of a confidence distribution and, correspondingly, confidence density 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. In this short note, the notion of statistically dual distributions is discussed. Based on properties of statistically dual distributions, a method for reconstructing the confidence density of a parameter is proposed.

Let

where

Let us name such distributions, which allow one to exchange the parameter and the variable, conserving the same formula for the distribution of probabilities, as statistically dual distributions [

In the next section, we show that Poisson and gamma distributions are statistically dual distributions and that the normal and Cauchy distributions are statistically self-dual distributions. An application of statistical duality for estimation of parameters is discussed in Section 3.

Definition: If a function

This definition is a purely probabilistic (and, in this sense, a frequentist) definition. Nevertheless, statistically dual distributions considered also belong to conjugate families defined in the Bayesian framework (see, for example, [

The statistical duality of Poisson and gamma distributions follows by a simple example. Let us consider the gamma distribution with pdf

Replacing

where a is a scale parameter and n + 1> 0 is a shape parameter. If a = 1 then the pdf of

The Poisson distribution is a popular model for counts. For instance, if there are n events of a certain kind then it is reasonable to say

One can see that the parameter and the variable in Equations (1) and (2) are exchanged. In other aspects the formulae are identical. As a result these distributions (gamma and Poisson) are statistically dual distributions. These distributions are connected by the identity [

for any

Another example of statistically dual distributions is the normal distribution with mean a and variance

where x is a real variable,

Hence, the normal distribution can be named as a statistically self-dual distribution. The identity analogous to (3) is

or, simply

for any real b, c and d.

The Cauchy distribution also has statistical self-duality like the normal distribution. The pdf of the Cauchy distribution is

where

Hence, the Cauchy distribution is also a statistically self-dual distribution. The identity analogous to (4) is

for any real

The same property applies to several other distributions, for example, the Laplace distribution.

The identity (3) can be written in form [

that is,

for any

The definition of the confidence interval

where

This definition is consistent with the identity (5). It contrasts with other frequentist definitions of confidence intervals. The right hand side of (6) represents the frequentist definition.

Let us suppose that

On the other hand: if

This identity is correct for any

If we subtract Equation (7) from Equation (5) then we have

We can choose

and hence a contradiction (i.e.

Statistically dual distributions allow one to exchange the parameter and the random variable. It means that one can construct the confidence interval

For the normal distribution the identity (4) can be written as

for any

This identity (8) also shows that the conditional distribution (if observed value is^{4}, taking into account systematics and statistical uncertainties in accordance with standard analysis of errors [

In case of the Cauchy distribution

for any

We have discussed the notion of statistically dual distributions. The relation between the measurement of a casual variable and estimation of the given distribution parameter is discussed for three pairs of statistically dual distributions.

The proposed approach allows one to construct the distribution of the estimator of the distribution parameter by using statistically dual distributions. For example, the confidence density of the Poisson distribution parameter can be built by Monte Carlo by using properties of statistically dual distributions [

In summary, statistical duality gives a clear frequentist interpretation of the “confidence density” of a parameter. It allows one to construct confidence intervals easily.

The authors are grateful to V. A. Kachanov, Louis Lyons, V. A. Matveev and V. F. Obraztsov for useful comments, R. D. Cousins, Yu. P. Gouz, G. Kahrimanis, V. Taperechkina and C. Wulz for fruitful discussions. This work has been particularly supported by the grant RFBR 13-02-00363.