Download REMARKS ON THE DEFINITION OF MODAL VALUE

REMARKS ON THE DEFINITION OF MODAL VALUE
Antoni Smoluk
Wroclaw University of Economics, Department of Mathematics
118 Komandorska Street
53-345 Wroclaw, Poland
asmoluk@manager.ae.wroc.pl
1. Modal value, a dominant, the most frequent value or simply a mode, is a positional
parameter informing about distribution of probability. In place of distribution of probability we
shall be using a synonymous expression, a probabilistic measure, shortened to one word, measure –
since only probabilistic measures will be mentioned.
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notion we should mention that not all the positional parameters are defined for each measure. There
are measures for which an average value is defined, and there are measures for which it is not so –
because it does not exist. Similarly this is the case with standard deviation and moments of higher
order. So for certain measures a modal value exists, and for others it does not. Before we propose a
new definition of a mode and attempt its generalization into multidimensional distribution let us
consult the literature. This notion does not figure in many of the textbooks on probability. Why?
Because it is not properly defined and there is not a precise definition of what a mode is. “The
mode is the value of the variable corresponding to the maximum of the ideal curve which gives the
closest possible fit to the actual distribution” (G.U.Yule, M. G. Kendall (1948)).
I think it is high time to review and make more precise this undoubtedly useful notion.
Statistics is not just a language and a method but above all a science of the real world, formulating
laws of nature.
2. Let be a probabilistic measure in a set of real numbers R. In a symbolic language it means
that ∈ Prob(R). Let x be a fixed real number. To the number x ∈ R and the measure ∈ Prob(R)
is assigned a function f( ,x) by the formula
0, when h < 0,

f ( µ , x)(h) = x + h
 ∫ dµ (t ), when 0 ≤ h,
 x −h
where h ∈ R, and an integral appearing in a definition of the function f( ,x) is the probability that
any random variable with distribution takes on values from the closed interval [x – h, x + h]. The
symbol F( ) designates a family of such functions F( ) = { f( [) : x ∈ R }.The family F( ) is an
order set: f( ,x) ”f( ,y) if and only if for every h ∈ R it is f( ,x) (h) ”f( ,y) (h).
If in the family F( ) exists a function f( ,m) such, that for every x ∈ R is f( ,x) ”f( ,m),
then the value m is called an ideal mode of the measure . The ideal mode of the measure exists
only then when in the family F( ) exists the largest element – supremum.
Theorem. An ideal mode is unique. If an ideal mode exists, it is only one because the
supremum of the set is unique. Every normal distribution has a mode and an ideal mode, both of
these values are equal the mean.
3. A mode for a uniform distribution is every point of an interval of positive density. In such a
case this notion has no value. However an ideal mode is unique, and obviously just like for normal
distribution, it equals the mean. The necessary condition for existing the ideal mode is symmetry of
distribution. The distribution has to be symmetrical in order to have an ideal mode. Naturally many
symmetrical distributions exist without an ideal mode. At the same time we should stress the great
importance of multimodal distribution in the analysis of pattern recognition and the theory of
classification. Multimodality cannot be fought against because the distributions of this kind take
place in natural and social sciences. However the notion of a mode has moderate usefulness in cases
when the distribution is multimodal. Simply - mode carries little information about distribution. An
ideal mode gives far more information than a mode. Full information about an ideal mode m is
contained in a distribuant f( ,m) defining an ideal mode m. Such distribuants carry also information
about an ordinary mode because each modal value m in a set of distribuant F( ), in a traditional
sense, corresponds with an element f( ,m). These modal values m which correspond with
distribuants f( ,m) deserve are maximum elements in a set F( ) deserve special attention as their
informative value is greater.
An ideal mode is a stable value due to a weak convergence of random variables.
Theorem. If a sequence of measures ( n) is weakly convergent to the measure and the
measure n for each natural n has an ideal mode mn, then the measure has an ideal mode m and
m = lim (mn).
The notion of an ideal mode can be easily extended onto multidimensional distributions.
4. Problems: A. Characterize a class of distributions for which an ideal mode exists.
Distribution possessing a density function has an ideal mode if and only if it is symmetrical and its
function of density to point of symmetry grows, and from the point of symmetry falls.
B. Does the definition of a mode not depend on the choice of a norm in a space Rn? Certainly
for normal distributions it really does not depend on the norm. If a measure has a central point –
ideal mode - in one norm, then in every other norm it has also a central point and all these points
are identical.
C. Multidimensional distribution has a centre m if and only if marginal distributions i,
i = 1, ..., n, have ideal modes mi, where m = (m1,..., mn)? It seems that it shall be so in case of the
independent marginal distributions, that is when is a product of marginal measures.
REFERENCES
Gellert, W. , Kästner, H. and Neuber, S. [Ed.] (1977). Lexikon der Mathematik. Bibliographisches
Institut. Leipzig.
Yule, G.U. and M. G. Kendall (1948). An Introduction to the Theory of Statistics. Charles Griffin
and Company. London.
REMARQUES SUR LA DÉFINITION DU MODE
Le mode est un concept simple et en raison de sa simplicité il est souvent utilisé. Cependant la
portée informatrice de ce paramètre est en général modeste. Dans ce travail, on rend plus précis le
concept de mode. Si dans une famille de fonctions de répartition F(µ), il existe une fonction f(µ, m)
telle que pour chaque x∈ R on a: f (µ , x ) ≤ f (µ , m ) la quantité m est nommée le mode idéal de
mesure µ. Le mode idéal de mesure µ existe si et seulement si dans une famille F(µ) il existe le
supremum. Le mode idéal est déterminé d’une façon univoque. Le concept de mode idéal est stable
en raison d’une faible convergence.
7KpRUqPH. Si la suite de mesures (µn) est faiblement convergente à une mesure µ, ainsi que la
mesure µn pour chaque n naturel a un mode idéal mn , la mesure µ a un mode idéal m et m = lim
(mn).