\documentstyle[IEEEtran]{article}
\begin{document}
\tolerance 10000
\title{Interval Methods For Processing Statistical
Characteristics}

\author{V. P. Kuznetsov}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\begin{abstract}
Traditional applications of interval computations are based on the
assumption that for all measured values, the only information that we
have about the error is the error bound $\Delta$ that is guaranteed by
the manufacturer of the measuring instrument. In such cases, the
only information that we have about the 
actual value $x$ of the measured quantity is that it belongs to an
{\it interval} $[\tilde x-\Delta,\tilde x+\Delta]$, where $\tilde x$ is the
result of the measurements.

In this paper, we consider the case when in addition to the
guaranteed error bound, we have an additional information about the
probabilities of different values of errors. We will show that
interval methods can be applied to this situation as well.
\end{abstract}

\auffil{The author is with the Dept. of Prognosis, 
Institute of Complex Systems, Moscow-Murmansk, Russia.
The author is thankful to V. Kreinovich 
for his help in writing and editing this text.}

\section{Where Do We Get Statistical Characteristics From?}

The information about these characteristics can come
from two possible sources:
\begin{itemize}

\item First, we may get it {\it theoretically}, if we know
what components the measuring instrument consists of, and how exactly
each component work (in particular, what are the probabilities of
different possible deviations of each component from its desired
behavior). 

\item In the majority of real-life cases, however, we do not
have that theoretical information. In these cases, the only source of
information about the error is {\it experiment}. 
\end{itemize}

How can we get an experimental value of error? The error is
defined as a difference $\tilde x-x$ between the measured and the actual
values of $x$. So, if we want to get the experimental value of an error,
we must somehow determine the actual value of the measured quantity $x$.
We cannot measure $x$ with an absolute
precision, because every measurement is inevitably
inaccurate. However, we can get a value that is (for our purposes)
indistinguishable from the actual value 
$x$ if we measure the same physical value $x$ by another measuring
instrument that is much more accurate than the one that we are testing.
This second measuring instrument is much more accurate and therefore,
much more expensive. For that reason, we do not use it for 
everyday measurements, we only use it as a {\it calibration
standard} to {\it calibrate} the measuring instrument that we will be
actually using. 

So, we arrive at the following experimental method of determining the
statistical characteristics of the measuring error:
We simultaneously measure the same
value of the physical quantity by two different measuring instrument:
\begin{itemize}

\item the one that we are testing (and that we will be
later on using for measurements), and

\item the {\it standard} measuring instrument that is
known to be practically perfect. 
\end{itemize}

Since we have chosen a (practically perfect) measuring instrument
as the second measuring instrument in this {\it calibration} procedure,
the value $\tilde x_{st}$ 
that is measured by this measurement instrument is practically
indistinguishable from the actual value: $\tilde x_{st}\approx x$. 
Therefore, the difference $\tilde x-\tilde x_{st}$ 
between the two measurement results is practically indistinguishable
from the error $\tilde x-x$ of this particular measurement. If we repeat
this comparative measurement several times, then we get several
different values of error. 

In terms of mathematical statistics, these values from a {\it
sample} of the unknown distribution. So, if we do not have any
theoretical ideas about the error distribution, then we have to extract
the information about the probabilities of different errors from this
sample. Depending on the size of the sample, we have two
different situations:
\begin{itemize}

\item If this sample is sufficiently
large (contains $\approx 10^3$ error values), then we can check whether
this sample is consistent with Gaussian or any other known family of 
probability distributions. If it is consistent with one of them, then we
know the family of distributions to which this particular error
distribution belongs, and the only thing that we still need to determine
from the sample are the parameters of this distribution. This case will
be considered in Section 1.3. 

\item If the sample is not so huge, then we do not have enough
values to determine the shape of the distribution, However, for
$\approx 100$ elements, we can determine the statistical characteristics
of this sample, such as the probability of the error to be in a certain
interval, moments of this distribution, etc. This case will be
considered in the current paper.
\end{itemize}

\section{Typical Statistical Characteristics}
Usually, as a first step of processing the sample
$v_1,...,v_N$ of error
values, we estimate the values of one or several of the following
statistical characteristics:
\begin{itemize}

\item {\it Moments}, i.e., characteristics of the type
$E(v^a)$ for a given integer $a$. 
\begin{itemize}
\item[]Here by $E$, we denoted {\it mathematical expectation} (i.e.,
$E(f(v))=\int f(v)\,d\mu$, where $\mu$ is the desired probability
measure). 
\item[]A moment is usually estimated simply as an arithmetic average
over the sample: $(v_1^a+...+v_N^a)/N$. 
\end{itemize}
\item{\it Generalized moments} of the type $E(f(v))$ for some
function $f(v)$ (e.g., $f(v)=|v|$, or $f(v)=(v-a)^2$ for some $a$).
\begin{itemize}
\item[]Just like regular moments, a generalized moment is estimated by an
arithmetic average $(f(v_1)+...+f(v_N))/N$.
\end{itemize}

\item {\it Probabilities} $p([a,b])$ that the error $v$
belongs to the given interval $[a,b]$. 
\begin{itemize}
\item[]This probability is estimated as the
ratio $M/N$, where $M$ is the number of values $v_i$ that lie inside the
interval $[a,b]$. 
\end{itemize}
\item []In particular, for $a=-\infty$, this probability
transforms into a {\it distribution function} $F(b)=p((-\infty,b])$. 

\item Another possibility is to measure the {\it inverse} to the
distribution function, i.e., given a probability $p\in [0,1]$, 
try to find $v$ for which $p((-\infty,v])=p$. The values of this
inverse characteristic are also called {\it quantiles}. 
If there are several such $v$, then we can take the smallest of them;
if the distribution function is discontinuous, then we can define the
quantile as $\sup\{x\,|\,p((-\infty,x]<x\}$.
\begin{itemize}
\item[]This characteristic is usually estimated 
by using the {\it ordinal} statistics,
i.e., the value $v_{(k)}$ that is $k-$th in increasing order among the
values $v_1,...,v_N$, and $k\approx p\cdot N$.
In particular, such characteristics include 
{\it maximum} $\max v_i$, {\it minimum} $\min v_i$, $\max |v_i|$, and
the {\it median} $v_{(N/2)}$.
\end{itemize}
\end{itemize}

\noindent{\it Comment.} 
The probability $p([a,b])$ can also be represented as a
generalized moment: namely, 
$p([a,b])=E(\chi_{[a,b]}(v))$ (where
$\chi_S(v)$ denotes a {\it characteristic function} of the set $S$,
i.e., a function that is equal to 1 if $v\in S$, and to 0 else). 
\smallskip

The characteristics obtained as a result of this processing 
are sometimes used to compute more complicated characteristics. For example:
\begin{itemize}
\item If we know the average $a=E(v)$, we can estimate the 
{\it mean square deviation} 
$E((v-a)^2)$.

\item If we know the median $a=v_{(N/2)}$, then we can compute
the {\it largest deviation} of the value from the median $\max|v_i-a|$.   
\end{itemize}

\noindent These characteristics, in their turn, can be used to
estimate further characteristics, etc. 

\section{For Each Statistical Characteristic, We Have An
Interval Of Possible Values}

\subsection{Statistical characteristics cannot be precisely
determined from experiments} 
The very idea of a random variable
include {\it randomness}, {\it impossibility to predict exactly.}
Because of that, it is impossible to precisely determine these characteristics
based on finitely many observations $v_i$, because no matter how we
process these observations, their random character will lead to an
unpredictable fluctuations of the processing result. 

\subsection{The more observations, the better the estimate}
Statistical characteristics are defined in terms of {\it probabilities} of
different values, and the probability of each event is defined as a {\it
limit} of the
frequency $M/N$, where $N$ is the total number of experiments, and $M$
is the number of experiments in which this event occurred. 
Therefore, the more observations we have, the closer the resulting 
estimate for probability to the actual value of this probability. 
Therefore, the estimate of a statistical characteristics (that is based
on these probability estimates) is getting closer to the actual value of this
characteristic as $N\to \infty$. How closer?

Let us answer this question characteristic-by-characteristic.

\subsection{The simplest case: estimating probabilities} 
Suppose that we want to determine the probability of a
certain event. To determine this probability, we repeat the experiment
$N$ times. If in these $N$ cases, the desired event occurred $M$ times,
then we take $M/N$ as the estimate for the probability. However, it is
not the actual probability. The larger $N$, the closer we are to the
actual probability. So, the possible actual values of the probability
form an {\it interval}. This interval is provided by mathematical statistics. 
(Actually, mathematical statistics provides us with {\it several}
different confidence intervals, but for simplicity, we will assume that
we have chosen one of them). 

\subsection{Moments and generalized moments}
Mathematical statistics also provides us with {\it intervals} of possible
values for moments and generalized moments.

\subsection{Ordinal statistics} 
For the ordinal statistics, the situation is slightly different, because
the inverse function to the probability distribution is not always
well defined: for
example, if we have a random variable that takes two values 0 and 1
with equal probability 0.5, then the median can be any number between
0 and 1, and therefore, its estimate $v_{(N/2)}$ either take the value
0, or the value 1. Therefore, no matter how many experiments we
undertake, these estimates alternate between 0 and 1 and never
converge to anything. 

To express the fact that for the increasing number of experiments $N$,
we do get a better estimate, we must consider the resulting estimate
$v_{(k)}$ {\it not} as an estimate for the {\it inverse} function to a
distribution function, but as an estimate for the {\it distribution
function itself.} Namely, we say that the number $p$ ($=k/N$); the
number that we started with) is an estimate for $p((-\infty,v_{(k)}))$.

\subsection{Extremal ordinal characteristics}
Extremal characteristics $v_{(1)}$ and $v_{(N)}$ (i.e., 
$\max v_i$ and $\min v_i$) have a different natural interpretation:
they are 
approximations for the interval of possible values of the random
variable. They can also be represented as estimates for generalized
moments: 
\begin{itemize}

\item If we knew {\it all} infinitely many 
values $v_1,...,v_N,...$, then we would be able to conclude that with
probability 1, the values of $v_i$ belong to the interval
$[\min(v_i),\max(v_i)]$. 

\item In real life, we only know {\it finitely many} values 
$v_1,...,v_N$. Therefore, we cannot guarantee that all the values
belong to the interval $[\min(v_i),\max(v_i)]$. However, we {\it can}
conclude that they belong to this interval with a certain probability
that tends to 0 as $N\to\infty$. 
\end{itemize}

The probability that a random number $v$ 
belongs to an interval $[a,b]$ can be also
described as a generalized moment: namely, as $E(\chi_{[a,b]})$. 
So, the accuracy of the extremal ordinal characteristics can also be
described by saying that the value of some generalized moment belongs
to a certain interval.

\subsection{Conclusion} 
In all cases, the estimates that we get from experiments are
interval estimates of the generalized moments, of the type 
$a^-_i\le E(f_i(v))\le a^+_i$.

\section{A General Description Of This Type Of Statistical
Information}

\noindent {\bf Definition 1.} \cite{Kuznetsov 1991} {\sl Let $X$ be a set. 
By a {\it (partial) statistical
information} (for a distribution defined on a set $X$), 
we mean a tuple $(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$, where:
\begin{itemize}

\item $m$ in a non-negative integer;

\item $f_i$, $1\le i\le m$, are functions from $X$ to $R$;
and

\item ${\bf a}_i$, $1\le i\le m$ are intervals.
\end{itemize}
We say that a probability measure $\mu$ on $X$ is {\it
consistent} with this information if for all $i$ from 1 to $m$, we
have $E(f_i)\in{\bf a}_i$ (where $E(y)$ denotes $\int y\,d\mu$).} 
\smallskip

\noindent{\it Comment 1.}
Since we are interested in probabilities of
errors, we will only consider the case when $X$ is either the set of
possible values of a single error (i.e., $X=R$), or $X$ is the set of all
possible values of all directly measured quantities (i.e., $X=R^n$ for
some $n\ge 1$).
\smallskip

\noindent{\it Comment 2.} 
In case of several variables, part of the information is
about individual variables, and part of it may be global. The
statistical information about the variable consists of the statements
of the type $E(f_i)\in {\bf a}_i$, where $f_i$ is a function of $x_i$.
Each of these statements can be easily described as a 
statement about the entire vector $\vec x(x_1,...,x_n)$: namely, as
$E(\tilde f_i)\in{\bf a}_i$, where $\tilde f_i$ is defined as 
$\tilde f_i(x_1,...,x_{i-1},x_i,x_{i+1},...,x_n)=f_i(x_i)$.
\smallskip

\noindent{\it Comment 3.}
In addition to the information coming from measurements,
we sometimes have an additional (a priori) information. In many cases, 
this additional (a priori) information can also be described in these
terms. For example, suppose that we know beforehand that the value of
the random variable $v$ always belongs to an interval $[a,b]$. This
information can be expressed as follows: the probability that $v\in [a,b]$
is equal to 1. If we express this probability in terms of a generalized
moment (as we have done in the previous subsection), then we can
express this condition as $E(f_i)\in {\bf a}_i$, where 
$f_i=\chi_{[a,b]}$ and ${\bf a}_i=[1,1]$. 

\section{Basic Problems of Processing Statistical
Information}

\subsection{Analysis} 
Before any data processing, 
the following question naturally arises \cite{Kuznetsov 1991}:
Suppose that we have some
partial statistical information, and we are interested in the value of
some other statistical characteristics. For example, we know the
second moment of the error, and we want to find the possible value of the
probability that this error belongs to a certain interval $[a^-,a^+]$.
Since the given information is {\it partial}, we do not expect a
precise number, but we would like to know the {\it interval} of
possible values of the desired characteristic.
\smallskip

\noindent{\bf Definition 2.} {\sl By the {\it problem of estimating the
value of a statistical characteristics}, we mean the following
problem:\\{\it Given:} 
\begin{itemize}
\item a set $X$;
\item a partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$ on a set $X$; and
\item a function $g:X\to R$.
\end{itemize}
{\it To compute:} the interval ${\bf a}=[a^-,a^+]$ 
of possible values of $E(g(x))$
for all distributions that are consistent with the given statistical
information.}

\subsection{Data processing} 
Informally, the problem of estimating
the errors of the result of data processing can be described as
follows:
\begin{itemize}
\item We have a partial statistical information about the
inputs $\vec x=(x_1,...,x_n)$. 
\item We know the function $y=f(x_1,...,x_n)$ that is used for
data processing.
\item We want to know the possible values of a certain
characteristics $g$ of $y$.
\end{itemize}
The desired characteristics can be expressed as
$E(y)=E(\tilde g)$, where we defined a new function 
$\tilde g:R^n\to R$ as $\tilde g(x_1,...,x_n)=g(f(x_1,...,x_n))$;
therefore, an arbitrary data processing problem can be represented as
a particular case of the analysis problem.   
 
\section{The General Methodology of Solving the Basic Problem Of
Statistical Data Processing: Main Idea}

In \cite{Kuznetsov 1991}, a methodology (based on linear programming) is
proposed that solves this problem. 
Linear programming is a conditional optimization problem in which 
the objective function
is linear in the variables, and the conditions that restrict the
values of the variables are linear inequalities. A standard
representation of a linear programming problem is in the following
form:
$$\sum_{j=1}^n c_jx_j\to\max$$ under the conditions that
$x_j\ge 0$ and for each $j$ from 1 to $n$, either 
$$\sum_{j=1}^m a_{ij}x_j\le b_i,$$ or
$$\sum_{j=1}^m a_{ij}x_j=b_i.$$ 
So, to apply linear
programming techniques, we must reformulate our problem in this form.
This will be done in three steps:
\begin{itemize}

\item First, we describe our problem as an optimization
problem. We need to find the interval of possible values of $E(g)$.
This means that we must find two values:
\begin{itemize}
\item[$\bullet$]the {\it largest} possible value of $E(g)$ for all
statistical distributions consistent with the given partial
information, and 

\item[$\bullet$]the {\it smallest} possible value of $E(g)$ for all
statistical distributions consistent with the given partial
information.
\end{itemize}
\item[]In other words, we must solve {\it two} optimization problems:
\begin{itemize}
\item[$\bullet$]$E(g)\to\max$ under the condition that $a^-_i\le
E(f_i)\le a^+_i$ for all $i$, and

\item[$\bullet$]$E(g)\to\min$ under the same conditions.
\end{itemize}
\item If $X$ is a finite set $X=\{a_1,...,a_n\}$, 
then, to define a probability measure, it is sufficient to define $n$
non-negative numbers $p_i=p(\{a_i\})$, $1\le i\le n$. These numbers
must satisfy the condition that the total probability is 1:
$p_1+...+p_n=1$. In terms of $p_i$, the value of the desired
characteristic $E(g)=\int g\,d\mu$ can be represented as $\sum
f(a_i)p_i$, i.e., a linear combination of the variables $p_i$.
Similarly, the conditions $a^-_i\le E(f_i)\le a^+_i$ can be
represented a linear inequalities. 

\item[]To get a similar representation for the general case (when the
set $X$ may be infinite), we can use the continuous analogue of the
probability at a point: namely, a probability density $\rho(x)$ with
respect to some standard measure (e.g., with respect to Lebesgue
measure for $X=R^d$). 
Therefore, instead of {\it finitely many} variables $p_i$, 
we will have {\it infinitely many} variables $\rho(x)$ that correspond
to all possible values $x\in X$. Another complication is that for some
probability measures (e.g., for a measure for which $x=0$ with
probability 1), 
$\rho(x)$ if a {\it generalized} function (i.e., a {\it distribution}
in the sense of functional analysis). However, we can still, {\it
formally}, 
write down our problem in a form that is linear in the ``variables''
$\rho(x)$:
$\int g(x) \rho(x)\,dx\to\max$ under the conditions that $\rho(x)\ge
0$ for all $x\in X$, $\int \rho(x)\,dx=1$, and $a_i^-\le \int
f_i(x)\rho(x)\,dx\le a^+_i$, $1\le i\le m$. 

\item This expression is almost in the desired form. The
only remaining step is to express the conditions in the standard form.
Namely, we need conditions of the type 
$\sum a_{ij}x_j\le b_i$ or
$\sum a_{ij}x_j=b_i$, while some of our conditions are 
inequalities of the opposite
sign: $\sum a_{ij}x_j\ge b_i$. 
In linear programming, it is well known how to reformulate these
conditions in a standard form: namely, we reformulate each of these
opposite-sign conditions as $\sum (-a_{ij})x_j\le (-b_i)$. 
\end{itemize}

As a result, we get the following representation of our
original problem: $\int g(x)\rho(x)\,dx\to\max$ under the conditions that 
$\rho(x)\ge 0$, 
$\int \rho(x)\,dx=1$, $\int f_i(x)\rho(x)\,dx\le a^+_i$, 
and $\int(-f_i(x))\rho(x)\,dx\le (-a^-_i)$. Here, we have:
\begin{itemize}
\item $x\in X$ instead of $j\in\{1,...,n\}$;

\item $\rho(x)$ instead of $x_j$; 

\item $g(x)$ instead of $c_j$;

\item 1, $f_i(x)$, and $-f_i(x)$ instead of $a_{ij}$;

\item 1, $a^+_i$, and $-a^-_i$ instead of $b_i$. 
\end{itemize}

Since we have a linear programming problem with {\it infinitely many
variables}, we cannot expect to apply known algorithms here. However,
we can apply the general tricks that helps to solve these problems.
One of the main tools in linear programming is the notion of {\it
duality}. Namely, according to duality theorem, the maximum attained
in the above-described linear programming problem coincides with the
minimum attained in the so-called {\it dual} problem. Namely, to each
of the conditions of the original problem, we assign the new variable
$y_i$, and define the dual problem as
$$\sum_{i=1}^n b_i y_i\to\min$$ under the conditions that
$y_i\ge 0$ for those conditions that were inequalities, and 
$$\sum_{i=1}^m a_{ij}y_i\ge c_j,\ 1\le j\le m.$$ Let us express this
formula in our case. Each of the new variables $y_i$ corresponds to
one of the conditions of the original linear programming problem. So, 
to make out notations clearer, let us use the following denotations:
\begin{itemize}

\item By $y_0$, we denote the
variable that correspond to the equality $\int\rho(x)\,dx=1$.

\item By $y^+_i$, we denote the variable that corresponds to
the inequality $\int f_i(x)\rho(x)\,dx\le a^+_i$.

\item By $y^-_i$, we denote the variable that corresponds to
the inequality $\int(-f_i(x))\rho(x)\,dx\le (-a^-_i)$. 
\end{itemize}

As a result, we get the following problem: $J^+\to\min$,
where we denoted
$$J^+=y_0+\sum_{i=1}^my^+_ia^+_i-\sum_{i=1}^my^-_i a^-_i,
\eqno{(1)}$$ under
the conditions that $$y^+_i\ge 0, y^-_i\ge 0,\eqno{(2a)}$$ and 
$$g(x)\le y_0+\sum_{i=1}^m y^+_i f_i(x)-\sum_{i=1}^m y^-_if_i(x) {\rm
\ for\ all\ } x\in X.\eqno{(2b)}$$ 
Similarly, to estimate the lower bound $a^-$ for
$E(g)$, we must solve the following problem: 
$J^-\to\max$, where 
$$J=y_0+\sum_{i=1}^my^+_ia^-_i-\sum_{i=1}^my^-_i a^+_i\eqno{(3)}$$ under
the conditions that $$y^+_i\ge 0, y^-_i\ge 0,\eqno{(4a)}$$ and 
$$g(x)\ge y_0+\sum_{i=1}^m y^+_i f_i(x)-\sum_{i=1}^m y^-_if_i(x) {\rm\ 
for\ all\ } x\in X.\eqno{(4b)}$$ 
So, we arrive at the following methodology:

\section{The General 
Methodology of Solving the Problems of Statistical Data Processing: 
Description}

\noindent{\bf Definition 3.} {\sl Suppose that a problem of estimating the
value of a statistical characteristic is given. By its {\it dual}, we 
mean the following two problems:
\begin{itemize}
\item the problem of minimizing the function (1) under
the conditions (2), and
\item the problem of maximizing the function (3) under
the conditions (4).
\end{itemize}
The solutions of these problems will be denoted
correspondingly by $m^-$ and $m^+$. The corresponding interval ${\bf
m}=[m^-,m^+]$ is called the {\it solution of a dual problem.}}
\smallskip

\noindent{\bf Definition 4.} {\sl We say that for a given problem of
estimating the value of a statistical characteristics, 
the {\it duality theorem} holds if the solution ${\bf a}$ to this
problem coincides with the solution ${\bf m}$ to its dual problem.}
\smallskip

\noindent{\it Comment 1.} Because of this definition, 
if the duality theorem holds, then one way
to find the bounds $a^\pm$ for $E(f)$ is to find the following two numbers:
\begin{itemize}

\item the minimum $m^+$ of the function (1) under the
conditions (2), and 

\item the maximum $m^-$ of the function (3) under the
conditions (4).
\end{itemize}
In the cases when duality theorem is true, these values
$m^\pm$ coincide with the desired values $a^\pm$. 
\smallskip

\noindent{\it Comment 2.} If
the duality theorem is not true, or if we do not know whether it is
true or not, computing $m^\pm$ still leads to 
reasonable estimates for $a^\pm$: 
\smallskip

\noindent{\bf PROPOSITION.} \cite{Kuznetsov 1991} 
{\it For every particular case of the problem of estimating the value
of a statistical characteristic, 
its desired solution ${\bf a}$ is a subset of the solution 
${\bf m}=[m^-,m^+]$ to a dual problem.}
\smallskip

So, we arrive at the following methodology:
\smallskip

\noindent{\bf METHODOLOGY 1.} (solving the basic problem of
statistical data processing)

{\bf Input:}
\begin{itemize}

\item {\tt a set $X$;}

\item {\tt a partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$ on a set $X$; and}

\item {\tt a function $g:X\to R$.}
\end{itemize}

{\bf Objective:} {\tt To estimate the interval ${\bf
a}=[a^-,a^+]$ of possible values of $E(g(x))$
for all distributions that are consistent with the given statistical
information.}

{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of possible values of the mathematical
expectation $E(g(x))$ for all
probability distributions for which $E(f_i(x))\in {\bf a}_i$ for
$i=1,...,m$.} 

{\bf Methodology:} 
\begin{itemize}

\item{\tt Solve the problem of minimizing the function
(1) under the condition (2); the resulting minimum will be
denoted by $m^-$.}

\item{\tt Solve the problem of maximizing the function
(3) under the condition (4); the resulting minimum will be
denoted by $m^+$.}

\item{\tt Return the interval $[m^-,m^+]$ as the desired
estimate.}
\end{itemize}

\noindent{\it Comment 1.}
If the duality theorem holds, then the resulting estimate
is precise. 
\smallskip

\noindent{\it Comment 2.} 
This Proposition reduces the solution of a problem to
the solution of its dual. In general, we do not know how to solve
the dual problem either, but in many important cases, the dual problem
is solvable. In these cases, we get a reasonable estimate for the
solution of the original problem. Several such cases are described in 
\cite{Kuznetsov 1991}:
\begin{itemize}

\item The case when 
we know the second moment, and we want to estimate the
probability of belonging to a given interval $[a,b]$. 
In this case, $g(x)=\chi_{[a,b]}(x)$, $m=1$, $f_1(x)=x^2$, and we are looking
for functions $y_0+y_1x^2$ with the property that $y_0+y_1 x^2\ge
\chi_{[a,b]}(x)$ for all $x$. This property can be explicitly
expressed in terms of $y_i$, and therefore, we get a simple and
solvable optimization problem. 

\item The case when we 
know the moment $E(|x|^\alpha)$ for some $\alpha>0$,
and we want to estimate $E(|x|^\beta)$ for some $\beta\ne\alpha$. 

\item The case when we 
know the generalized moments $E(\exp(-k|x|)$, and
we want to estimate probabilities $p([a,b])$. 

\item The case when we 
know moments, and we want to estimate Fourier
transforms $E(\sin(\omega x))$ and $E(\cos(\omega x))$.
\end{itemize}

3. In some cases, we do not know how to solve the dual
problem either. In these case, we can still use this Proposition to 
get some estimates for ${\bf a}$. This possibility is based on the
following facts: 
\begin{itemize}

\item according to our Proposition, $a^-\ge m^-$;

\item $m^-$ is defined as the maximum of the function $J^-$
(defined by the formula (3)) for
all $y^\pm_i$ that satisfy the condition (4); therefore, if for
some $y^\pm_i$, we get the conditions (4), then the corresponding
value of $J^-$ is $\le m^-$ and therefore, $\le a^-$. 
\end{itemize}

A similar estimate can be done for $a^+$, so, we get the
following corollary:
\smallskip

\noindent{\bf COROLLARY.} {\it Assume that a problem of
estimating the value of a statistical characteristics is given. Then,
the solution ${\bf a}=[a^-,a^+]$ of this problem satisfies the
following properties:}
\begin{itemize}

\item {\it If the numbers $y_0$ and $y^\pm_i$ satisfy the
condition (2), then $J^+(y_0,\{y^\pm_i\})\ge a^+$.} 

\item{\it If the numbers $y_0$ and $y^\pm_i$ satisfy the
condition (4), then $J^-(y_0,\{y^\pm_i\})\le a^-$.} 
\end{itemize}

\noindent{\bf METHODOLOGY 2.} (solving the basic problem of
statistical data processing)

{\bf Input:}
\begin{itemize}

\item {\tt a set $X$;}

\item {\tt a partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$ on a set $X$; and}

\item{\tt a function $g:X\to R$.}
\end{itemize}

{\bf Objective:} {\tt To estimate the interval ${\bf
a}=[a^-,a^+]$ of possible values of $E(g(x))$
for all distributions that are consistent with the given statistical
information.}

{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of possible values of the mathematical
expectation $E(g(x))$ for all
probability distributions for which $E(f_i(x))\in {\bf a}_i$ for
$i=1,...,m$.} 

{\bf Methodology:} 
\begin{itemize}

\item {\tt Find  one or several tuples 
$(y_0, y^\pm_1,...)$ for which the
inequalities (2) are true; for each of these tuples, compute the
value $J^+$ by using the formula (1). Denote the smallest of the
resulting values by $m^-$.}

\item {\tt Find  one or several tuples 
$(y_0, y^\pm_1,...)$ for which the
inequalities (4) are true; for each of these tuples, compute the
value $J^-$ by using the formula (3). Denote the largest of the
resulting values by $m^+$.}

\item {\tt Return the interval $[m^-,m^+]$ as the desired
estimate.}
\end{itemize}

\section{Proof of the Proposition}

The proof is reasonably simple: Suppose that the conditions (2) are true. 
Then, the difference 
$$d(x)=y_0+\sum y^+_i f_i(x)-\sum y^-_if_i(x)-g(x)$$ is non-negative.
Therefore, its mathematical expectation $E(d)$ is also non-negative.
Hence, $$E(g)\le E(y_0+\sum y^+_i f_i(x)-\sum y^-_if_i(x))=$$ $$y_0+\sum
y^+_i E(f_i)-\sum y^-_i E(f_i).$$ From $a^-_i\le E(f_i)\le a^+_i$, it
follows that $E(g)\le J^+$. Hence, $E(g)\le \inf J^+=m^+$. Since this
is true for every measure that is consistent with the given
information, we conclude that $a^+=\sup E(g)\le m^+$. Similarly, one
can prove that $a^-\ge m^-$. Q.E.D. 

\begin{thebibliography}{99}
\bibitem{Kuznetsov 1991} 
V. P. Kuznetsov, {\it Interval statistical models}, Moscow, Radio i
Svyaz Publ., 1991 (in Russian). 
\end{thebibliography}
\end{document}

