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\title{Invariance Leads to the Interval Character of Ordinal
Statistical Characteristics}

\author{A. I. Orlov}

\pagestyle{myheadings}

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

\maketitle

\auffil{The author wants to thank V. Kreinovich for his help in
writing and editing this
text.}

Traditionally, in measurement theory, 
a unit of measurement is fixed, 
and the starting point determine the value of the
quantity uniquely. However, in many real-life cases, we also have 
to describe quantities for which only the {\it order} makes sense: e.g.,
when a student is stronger than another, when a profession is more
attractive than another, etc. In some of these cases, there may be a way
to ascribe a meaningful numerical value to each degree of
attractiveness, but, if such a method is not known, we have to be
consent with the estimates for which only the order matters, i.e., which
are defined modulo an arbitrary monotonic transformation. 

In such cases, if we have several measurement results, we would like to
describe some statistical characteristics. For these statistical
characteristics to be meaningful, it is necessary to demand that they
are invariant with respect to arbitrary monotonic transformations.
This exclude such characteristics as, e.g., average, because average
is not invariant: e.g., if in one scale $x$, 
we have three measurements 1, 2, and 3, then
the average is 2. If we now use a different scale $y=2^x$, the same
measurements will be equal to 2, 4, and 8. If the average was invariant,
then we would have the average in the new scale equal to $2^2=4$, but
the actual average of these three values is $14/3\ne 4$. 

How to describe all possible numerical 
statistical characteristics for this case?
By a {\it characteristic}, we mean a way to transform $N$ given measurement
results into a single value that is invariant with respect to arbitrary
permutations (because the order of measurements is irrelevant), and
arbitrary monotonic rescalings. The third reasonable demand is based on
the fact that the measurements on which we base our characteristics may
not be absolutely precise. Therefore, we base our computations of this
characteristic on the {\it approximate} values of the measured quantity.
It is therefore reasonable to demand that when we apply more and more
precise measurements, the resulting value of the characteristic will be
more and more accurate. In other words, it is reasonable to demand that
the characteristic is expressed by a continuous function. So, we arrive
at the following definition:
\smallskip

\noindent{\bf Definition 1.} {\sl By an {\it ordinal numerical statistical
characteristic}, we mean a continuous function $f:R^n\to R$ that
satisfies the following two properties:
\begin{itemize}

\item It is 
invariant with respect to permutations, i.e.,
$f(x_{\pi(1)},...,x_{\pi(N)})=f(x_1,...,x_N)$ for an arbitrary
permutation $$\pi:\{1,2,...,N\}\to\{1,...,N\}.$$

\item It is invariant with respect to arbitrary monotonic
rescalings $\phi$, i.e.,
$f(\phi(x_1),...,\phi(x_N))=\phi(f(x_1,...,x_N))$ for an arbitrary
monotonically increasing 1--1 function $\phi:R\to R$.
\end{itemize}}

The description of all possible ordinal numerical statistical
characteristics is basically, 
given in \cite{Orlov 1974} (we give a slightly more
general statement here):
\smallskip

\noindent{\bf PROPOSITION 1.} 
{\it Every ordinal numerical statistical
characteristic is of the type $x_{(k)}$ for some $k$.}
\smallskip

We have already mentioned that from finitely many experiments, we can
only make an {\it approximate estimate} of the desired statistical
characteristic. As a result of such an approximate estimation, we will
have not a single numerical value, but the {\it interval} of possible
values. How to describe the resulting {\it interval} characteristics? 
\smallskip

\noindent{\bf Definition 2.} {\sl By an {\it ordinal interval statistical
characteristic}, we mean a continuous function $\bf f$ from $R^n$ into
the set of all intervals that
satisfies the following two properties:
\begin{itemize}

\item It is 
invariant with respect to permutations, i.e.,
${\bf f}(x_{\pi(1)},...,x_{\pi(N)})={\bf f}(x_1,...,x_N)$ for an arbitrary
permutation $$\pi:\{1,2,...,N\}\to\{1,...,N\}.$$

\item It is invariant with respect to arbitrary monotonic
rescalings $\phi$, i.e.,
${\bf f}(\phi(x_1),...,\phi(x_N))=\phi({\bf f}(x_1,...,x_N))$ for an arbitrary
monotonically increasing 1-1 function $\phi:R\to R$.
\end{itemize}}

\noindent{\bf PROPOSITION 2.} 
{\it Every ordinal interval 
characteristic is of the type $[x_{(k)},x_{(l)}]$ for some integers
$k\le l$.}
\smallskip

\noindent{\it Comment.}
These propositions describes all possible ordinal characteristics,
i.e., characteristics that are defined modulo arbitrary monotonic
rescalings. For these characteristics, intervals appear because we are
trying to estimate the value of a statistical characteristic based on
finitely many measurements. There is a
special case when a numerical characteristics is simply impossible: the
case when we try to estimate the {\it average} value of a sample (in
some reasonable sense). A natural property of an ``average'' value is
that that if we invert the order, e.g., change $x_i$ to $-x_i$, then
the average must also change its sign. So, we arrive at the following
definition:
\smallskip

\noindent{\bf Definition 3.} {\sl 
\begin{itemize}

\item An ordinal numerical characteristic $$f:R^N\to R$$ is
called an {\it average} if for all $x_i$, 
$$f(-x_1,...,-x_N)=-f(x_1,...,x_N).$$
\item An ordinal interval characteristic $${\bf f}:R^N\to {\bf I}$$
is called an {\it average} if for all $x_i$,
$${\bf f}(-x_1,...,-x_N)=-{\bf f}(x_1,...,x_N).$$ 
\end{itemize}}

\noindent{\bf PROPOSITION 3.} 
\begin{itemize}

\item {\it If $N$ is even, then there exists only one average
ordinal numerical characteristic: the median $x_{(N/2)}$.} 

\item {\it If $N$ is odd, then there exist no average
ordinal numerical characteristics.}

\item {\it Every average ordinal interval characteristic has
the form $[x_{(k)},x_{(N-k)}]$ for some $k\le N/2$.}
\end{itemize}

\noindent{\it Comment.} If $N$ is even, then the narrowest possible
estimate is a numerical estimate, namely, the median. What is the
narrowest possible average for odd $N$? The answer to this question
easily follows from the above proposition:
\smallskip

\noindent{\bf Definition 4.} 
{\sl An average ordinal interval characteristic ${\bf f}(x_1,...,x_n)$ is
called an (interval) {\it median} if it is the narrowest of all possible
average ordinal interval characteristic ${\bf g}(x_1,...,x_n)$ (i.e.,
if ${\bf f}(x_1,...,x_n)\subseteq {\bf g}(x_1,...,x_n)$ for all $x_i$).} 
\newpage

\noindent{\bf PROPOSITION 4.} {\it For every $N$, there is exactly one
interval median $\bf f$:}
\begin{itemize}

\item {\it When $N$ is even, then $${\bf
f}(x_1,...,x_n)=[x_{(N/2)},x_{(N/2)}].$$}

\item {\it When $N$ is odd, then $${\bf
f}(x_1,...,x_n)=[x_{((N-1)/2)},x_{((N+1)/2)}].$$}
\end{itemize}

\noindent{\it Comment 1.}
The lower and upper bounds of the interval median are
called {\it left} and {\it right} medians. 
\smallskip

\noindent {\it Comment 2.} 
For odd $N$, {\it there is no numerical characteristic that
expresses the notion of the ``average'', only an interval one}. 

\subsection{Applications}

This notion has been applied to the following areas:

\subsubsection{Case study: sociology} Students from high schools of two
different cities were asked to estimate their preference of different
professions on a scale. The results from \cite{Shubkin 1970,Shchukina 1971}
were analyzed first by using traditional statistical methods (average),
which lead to a conclusion that the average preferences differed from
city to city. However, since preferences are defined only modulo an
arbitrary monotonic transformation, applying an arithmetic average makes
no big sense. When interval medians were applied, the differences
between high school students from different cities turned out to be
insignificant \cite{Orlov 1993}. 

\subsubsection{Case study: quality measurements and quality control}
In many cases, there is no objective measure of quality, so, we have to
ask experts to estimate quality on a scale (from, say, 0 to 10). Then,
if we want to compare the average quality assigned by experts, we have
to use interval medians to describe these averages 
\cite{Krivtsov 1988,Orlov 1993}. 

\subsection{Proofs}

\subsubsection{Proof of Proposition 1.} Indeed, let $x_i=i$. Let us choose
the following function $\phi$: $2x-1$ for $x<1$, $2x-N$ for $x>N$,
$\phi(x)=i+1.5(x-i)$ when $i\le x\le i+0.5$, $i=1,...,N-1$, and
$\phi(x)=i+0.75+0.5(x-i-0.5)$ when $i+0.5\le x\le i+1$. This function is
strictly increasing, 1--1, and $\phi(x)=x$ only for $x=1,2,...,N$.
Therefore, from the condition that
$f(\phi(1),...,\phi(N))=\phi(f(1,...,N))$, we conclude that
$f(1,...,N)=\phi(f(1,2,...,N))$ and therefore, that $f(1,...,N)=k$ for
an integer $k\in \{1,2,...,N\}$. 

If the values $x_i$ are all different, then, due to permutation
invariance, we get $f(x_1,...,x_N)=f(x_{(1)},...,x_{(N)})$. One can
easily construct a 1--1 monotonic transformation $\phi(x)$ that maps $i$ into
$x_{(i)}$ and therefore,
$f(x_1,...,x_N)=
f(x_{(1)},...,x_{(N)})=f(\phi(1),...,\phi(N))=
\phi(f(1,...,N))=\phi(k)=x_{(k)}$.
So, for the case when all $x_i$ are different, we get the desired
equality. Cases when some of the values $x_i$ coincide can be easily
obtained from these ones by tending to a limit. So, since we assumed
that $f$ is continuous, this expression is true for all $x_i$. Q.E.D.

\subsubsection{Proof of Proposition 2.} Indeed, it is easy to see that 
if ${\bf f}=[f^-,f^+]$ is
an ordinal interval characteristic, then both $f^-$ and $f^+$ are 
ordinal numerical characteristics, and thus, due to the previous
Proposition, each of them can be described as $x_{(k)}$ for some $k$.
Q.E.D.

\subsubsection{Proof of Proposition 3.} This proposition follows from the
fact that for $y_i=-x_i$, the ordering of $y_{(i)}$ is opposite to the
ordering of $x_{(i)}$, i.e., $y_{(i)}=-x_{(N-i)}$. So, for interval
characteristic ${\bf f}(x_1,...,x_n)=[x_{(k)},x_{(l)}]$, the condition
that $\bf f$ is an average means that $[x_{(k)},x_{(l)}]=
{\bf f}(x_1,...,x_n)=-{\bf f}(-x_1,...,-x_n)=
-[y_{(k)},y_{(l)}]=-[-x_{(N-k)},-x_{(N-l)}]=[x_{(N-l)},x_{(N-l)}]$. 
Therefore, $x_{(k)}=x_{(N-l)}$, and $l=N-k$. The statement about
numerical characteristics follows if we take $k=l$. Q.E.D.

\begin{thebibliography}{99}
\bibitem{Krivtsov 1988}
V. S. Krivtsov, V. N. Fomin, and A. I. Orlov, ``Modern statistical
methods in standardization and quality control'', {\it Standards and Quality},
1988, No. 3, pp. 32--36 (in Russian). 

\bibitem{Orlov 1974}
A. I. Orlov, ``Admissible averages in the problems of expert
estimation and agregation of quality characteristics'', In: 
{\bf Multidimensional statsitical analysis in socio-economic research},
Moscow, Nauka Publ., 1974, pp. 388--393 (in Russian). 

\bibitem{Orlov 1993}
A. I. Orlov, ``On the development of the statistics of nonumeric
objects'', In: E. K. Letzky (ed.), {\bf Design of experiments and data
analysis: new trends and results}, Antal, Moscow, 1993, pp. 53--90.

\bibitem{Shchukina 1971}
G. I. Shchukina, {\bf Problems of cognitive interest in pegagogics},
Moscow, Pedagogica Publ., 1971 (in Russian). 

\bibitem{Shubkin 1970}
V. N. Shubkin, {\bf Sociological experiments}, Moscow, Mysl Publ.,
1970 (in Russian).

\end{thebibliography}
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