In this paper a set of desirable properties for measures of positive dependence of ordered n-tuples of continuous random variables (n >= 2) is proposed and a class of multivariate positive dependence measures is introduced. We consider the comonotonicity dependence structure as the strong dependency structure and so the class consists of measures that take values in the range [0, 1] and are defined in such a way that they equal 1 in case the random vector is comonotonic and equal 0 in case it is independent.

Marta Cardin

English

Research Papers in Economics

Department of Applied Mathematics, University of Venice 
 
 
WORKING PAPER SERIES 
 
 
Working Paper n. 166/2008 
September 2008 
 
ISSN: 1828-6887 
 
 
 
 
 
 
 
 
Marta Cardin 
 
 
Multivariate measures of positive dependence 
 
 
 
 
 
 
 
 
 This Working Paper is published under the auspices of the Department of Applied Mathematics of 
the Ca’ Foscari University of Venice. Opinions expressed herein are those of the authors and not 
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consider its provisional nature. 
 
 
 
 
 
 
 
 Multivariate measures of positive dependence
Marta Cardin
<mcardin@unive.it>
Dept. of Applied Mathematics
University of Venice
Abstract. In this paper a set of desirable properties for measures of positive dependence
of ordered n-tuples of continuous random variables (n ¸ 2) is proposed and a class of
multivariate positive dependence measures is introduced. We consider the comonotonicity
dependence structure as the strong dependency structure and so the class consists of mea-
sures that take values in the range [0;1] and are de¯ned in such a way that they equal 1 in
case the random vector is comonotonic and equal 0 in case it is independent.
Keywords: Measures of dependence, copula, comonotonicity.
JEL Classi¯cation Numbers: 62H05, 62H20, 60E15.
Correspondence to:
Marta Cardin Dept. of Applied Mathematics, University of Venice
Dorsoduro 3825/e
30123 Venezia, Italy
Phone: [++39] (041)-234-6929
Fax: [++39] (041)-522-1756
E-mail: mcardin@unive.it
1Introduction
The concept of dependency in a bivariate and in a multivariate setting has been largely
studied in recent literature and the development of the theory of copulas has determined
a great impact in the study of measures of dependence. This is of interest in economics,
insurance, in ¯nance and risk management and in many areas of applied probability and
statistics. Several notions of positive dependence were introduced in the literature to model
the fact that large values of a component of a multivariate random vector are \probabili-
stically associated" with large values of the others. As it is well known that the most widely
applied dependence measures, the Pearson's product moment correlation coe±cient which
captures the linear dependence between couples of random variables is a weak measure in
many setups and consider only pairwise dependency. So there are a variety of ways to
consider and to measure dependence between random variables and this fact con¯rm the
importance and the high interest in the concept. For a review see e.g [1], [4] and[5]. These
concepts have been introduced on the class of bivariate random vectors and many of these
orders can be further extended to comparison of general multivariate distributions that have
the same marginals.
The focus of this paper is on a class of measures of multivariate positive dependence de¯ned
by means of some comparison between a joint distribution function and a distribution
representing independence. Properties of these measures are also investigated in a axiomatic
framework.
The paper is organized as follows. In Section 2, notations will be ¯xed and various concepts
and results that are essential to the development of the present paper are reviewed. After
a brief overview about some properties of copula, we introduce the copula approach for
studying dependence between random variables and Section 3 presents the main results.
1 Notations and preliminaries
We are going to review some basic de¯nitions and properties about dependence concepts
and dependence orders which we will use later. We will be concerned with random vectors
that take on values in Rn. Elements of Rn will be denoted by x;y or more explicitly, as
x = (x1;:::;xn) or y = (y1;:::;yn).
For two n-vectors x and y, the notation x · y will be used for the componentwise order
which is de¯ned by xi · yi for all i = 1;:::;n.
Throughout this paper, all the random variables considered are de¯ned on a common prob-
ability space (­;F;P). The set ­ consists of all possible results or outcomes and its generic
element is denoted by ! and the ¾-¯eld F is a collection of subset of ­.
The distribution function of a random vector X = (X1;:::;Xn)) is the function F : Rn !
[0;1] de¯ned as follows:
F(x) = P [X1 · x1;:::;Xn · xn]
A distribution function is non-decreasing and right-continuous on Rn while the survival
function is a non-increasing, right-continuous function so de¯ned:
2¹ F(x) = P [X1 > x1;:::;Xn > xn]
If we want to consider the distribution of Xi;i = 1;:::;n singularly, we need the marginal
distribution function Fi (xi) = P [X1 · x1]:
Fr¶ echet spaces are important in studying dependence between random vectors, paying at-
tention to probability distributions with ¯xed univariate marginals and so the elements of a
Fr¶ echet space only di®er in their dependence structure and not in the marginals behaviors.
If F1;:::;Fn are univariate distribution functions the Fr¶ echet space R(F1;:::;Fn) consists
of all the n-dimensional random vectors with F1;:::;Fn as marginal distributions.
Some random variables X1;:::;Xn are mutually independent if and only if
F (x) =
Qn
i=1 Fi (xi);
where x 2 Rn and F is the distribution function of the random vector X = (X1;:::;Xn).
In particular if X is a random vector we denote by X? the random vector whose marginal
distributions coincide with the marginals of X, but whose components are independent.
Now we introduce the concept of comonotonicity by de¯ning comonotonicity of a set of
vectors in Rn.
De¯nition 1.1 The set A ½ Rn is comonotonic if for any x;y in A, either x · y or y · x
holds.
So a set A µ Rn is comonotonic if for any x;y in A, the inequality xi < yi for some i, implies
that x · y. Next we de¯ne a comonotonic random vector X through its support where a
support of a random vector X is the smallest closed set A µ Rn for which P [X 2 A] = 1.
De¯nition 1.2 The random vector X is comonotonic if it has a comonotonic support.
The following result provides some equivalent conditions of comonotonicity.
Proposition 1.1 If X = (X1;:::;Xn) is a n-dimensional random vector the following
statements are equivalent:
i) The random vector X is comonotonic;
ii) For any i;j with 1 · i;j · n the inequality
[Xi(!1) ¡ Xi(!2)][Xj(!1) ¡ Xj(!2)] ¸ 0
holds almost surely for !1 and !2 in ­;
iii) If F is the distribution function of the random vector X for all x = (x1;:::;xn) we
have F (x) = minfF1 (x1);:::;Fn (xn)g;
iv) There exists a random variable Z and n increasing functions ti, we obtain X =d
(t1 (Z);:::;tn (Z)).
3The symbol \=d"in the above result means \is equally distributed as".
So in order to ¯nd the probability of all the outcomes of n comonotonic vectors being less
than xi, one simply takes the probability of the least likely of these n events. Comonotonicity
is indeed a very strong positive dependency structure. A possibility to introduce some
weakest form of positive dependence dependence concept is to consider the set of all random
vectors which are larger than X? with respect to some dependence order. We consider in
particular the positive quadrant dependence orders.
De¯nition 1.3 Let X;Y are two n-dimensional random vectors with distribution functions
F and G and survival functions ¹ F and ¹ G respectively. We say that X is smaller then Y in
the positive lower orthant dependence order and we denote
X¹puodY , F(t) · G(t); for all t 2 Rn
and we say that X is smaller then Y in the positive upper orthant dependence order and
we denote
X¹plodY , ¹ F(t) · ¹ G(t); for all t 2 Rn
The positive orthant orders measure the amount of positive dependence of the underlying
random vectors. Now we consider some notions of positive dependence for multivariate
random vectors.
De¯nition 1.4 If X a n-dimensional random vector we say that X is a positive lower
orthant dependent(PLOD) if X¸plodX? and is a positive upper orthant dependent (PUOD)
if X¸puodX?.
In this paper we use the term \positive dependence " for a random vector which is positive
lower orthant dependent. Then if X is a n-dimensional random vector in R(F1;:::;Fn)
with distribution function F, X is positive dependent if and only if
F (x) ¸
n Y
i=1
Fi (xi)
We denote by FI the distribution function of the random vector in R(F1;:::;Fn) with
independent components,
FI (x) =
n Y
i=1
Fi (xi) (1)
and FC the distribution function of the random vector in R(F1;:::;Fn) with comonotonic
components.
FC (x) = minfFX1 (x1);:::;FXn (xn)g (2)
Then for PLOD random vectors it is possible to prove the following result:
Theorem 1.1 If X be a PLOD n-dimensional random vector in R(F1;:::;Fn) with distri-
bution functions F then for all x 2 Rn
FI (x) · F (x) · FC (x):
4The concept of copula introduced by Sklar in 1959 is now common in the statistical lit-
erature, but only recently its potential for applications has become clear. Copulas permit
to represent joint distribution functions by splitting the marginal behavior, embedded in
the marginal distributions, from the dependence captured by the copula itself. A copula
can also be seen as a joint probability distribution with uniform marginals. Elimination of
marginals through copulas helps to model and understand dependence structure between
variables more e®ectively, as the dependence has nothing to do with the marginal behav-
ior. First of all we are focusing our attention to the n-increasing property for multivariate
functions.
If f is a function Rn ! R, a B is the n-box B = [a1;b1] £ ::: £ [an;bn] µ Rn with ai < bi,
i = 1;:::;n, then the f-volume of B is
Vf(B) =
2 X
i1=1
:::
2 X
in=1
(¡1)i1+:::+inf(x1i1;:::;xnin)
where xi1 = ai, xi2 = bi.
If there exists i 2 1;:::;n such that ai = bi then Vf(B) = 0.
Equivalently the f-volume of B is the n-order di®erence of f on B is :
Vf(B) = ¢(a1:::;an);(b1:::;bn)f(x1;:::;xn) = ¢a1;b1 :::¢an;bnf(x1;:::;xn)
where we de¯ne the n ¯rst order di®erences of f as
¢ak;bkf(x1;:::;xn) = f(x1 :::xk¡1;bk;xk+1;:::;xn) ¡ f(x1 :::xk¡1;ak;xk+1;:::;xn):
Moreover by convention if k = n we set
¢(a1:::;an);(b1:::;bn)f(x1;:::;xn) = f(x1;:::;xn):
De¯nition 1.5 A function f: Rn ! R is n-increasing if for all n-box B = [a1;b1] £ ::: £
[an;bn] µ [0;1]n where ai · bi;i = 1;:::;n, Vf(B) ¸ 0.
If f has nth-order derivatives, n-increasing is equivalent to @n
@x1:::@xnf ¸ 0.
De¯nition 1.6 A n-copula is a function C : [0;1]n ! [0;1] that satis¯es:
(a) C(x1;:::;xn) = 0 if xi = 0 for any i = 1;:::;n.
(b) C(1;:::;1;xi;1;:::;1) = xi for each i = 1;:::;n and all xi 2 [0;1]:
(c) C is n-increasing.
Conditions (a) and (b) are known as boundary conditions whereas condition (c) is known as
monotonicity. In fact the de¯nition of n- increasing function is the multivariate extension
of the concept of \increasing" for a univariate function when we interpret \increasing" as
\increasing as a distribution function".
5Various properties of copulas have been studied in literature, but most part of the research
concentrates on the bivariate case, since multivariate extensions are generally not easily to
be done.
The following theorem (Sklar 1959), which partially explains the importance of copulas in
statistical modeling, justi¯es the role of copulas as dependence functions
Theorem 1.2 (Sklar's Theorem) Let F be an n-dimensional distribution function with
marginal distribution F1;:::;Fn. Then there exists an n-dimensional copula C(u1;:::;un)
such that
F(x1;:::;xn) = C(F1(x1);:::;Fn(xn))
for all x = (x1;:::;xn) 2 Rn.
If the functions F1;:::;Fn are all continuous then C is unique. Conversely if C is a n-
dimensional copula and F1;:::;Fn are distribution functions, then the function F de¯ned
above is a n-dimensional distribution function with marginal distribution F1;:::;Fn.
A rigorous mathematical description of copulas and their features is available in [8].
If we consider the independent copula
Ci(u1;:::;un) = u1u2;:::;un
and the comonotonic copula
Cc(u1;:::;un) = minfu1;u2;:::;ung
and we call a n-copula C a PLOD copula if
C(u1;:::;un) ¸ Ci(u1;:::;un)
for all (u1;:::;un) 2 Rn it is easy to prove that if C is a n-dimensional copula
Ci(u1;:::;un) · C(u1;:::;un) · Cc(u1;:::;un)
for all (u1;:::;un) 2 Rn.
If µ is the permutation (1;:::;n) 7! (i1;:::;in) and C is a n-dimensional copula we denote
by Cµ the copula de¯ned by C(u1;:::;un) = C(ui1;:::;uin).
2 Main Results
The purpose of this paper is to investigate the notion of dependence between multidi-
mensional distributions with the same marginal distributions by proposing some complete
dependence orderings. In particular Joe de¯ned a set of axioms that a dependence ordering
of distributions should have in order that higher in the ordering means more positive de-
pendence. These properties may be generalized to cover the case of n-variate distribution
6functions. We suggest a list of desirable properties that any multivariate positive depen-
dence notion should ful¯ll.
By a measures of positive dependence we means a function that attaches to every n-tuple of
continuous and PQD random variable X = (X1;:::;Xn) where n ¸ 2, a real number m(X)
satisfying the following:
P1. (Normalization) m(X)=0 if and only if and X has independent component and
m(X)=1 if and only the random vector X is comonotonic;
P2. (Monotonicity) If X¹plod Y then m(X) · m(Y);
P3 (Permutation Invariance) m(Xi1;Xi2;:::;Xin) = m(X1;X2;:::;Xn)
for all permutations i1;:::;in of f1;:::;ng.
P4 (Functional Invariance) m((X1;X2;:::;Xn)) = m((a(X1);X2;:::;Xn)) for all in-
creasing continuous function a;
P5 (Continuity) If Xn ! X in distribution then m(Xn) ! m(X).
The following result show that the copula accounts for all the dependence between random
variables.
Theorem 2.1 Let X = (X1;:::;Xn) be an n-dimensional random vector with copula C and
a1;:::;an be increasing and continuous functions. Then the random vector (a1(X1);:::;an(Xn))
has the copula C.
Invariance properties of copulas suggests that they facilitate the study of scale-free mea-
sures of dependence and so we reformulate the axioms that characterize measures of positive
dependence in terms of copulas.
If n ¸ 2, we indicate by C+(n) the set of PLOD n-dimensional copulas .
By a measures of positive dependence we means a function m : C+(n) 7! R satisfying the
following:
A1. m(C)=0 if and only if C = Ci and m(C)=1 if and only C = Cc;
A2. If C1 · C2 pointwise then m(C1) · m(C2);
A3 m(Cµ) = m(C) for all permutations µ of f1;:::;ng;
A4 If Cn ! C pointwise then m(Cn) ! m(C).
If µ is the permutation (1;:::;n) 7! (i1;:::;in) we denote by fµ the function de¯ned by
fµ(x1;:::;xn) = f(xi1;:::;xin).
De¯nition 2.1 A Borel measure ¹ is permutation invariant if for any Borel set B and any
permutation µ, ¹(fµ(B)) = ¹(B).
7In this contribution the strength of dependence between n variables is compared using the
notion of comonotonicity and we consider measures of positive dependence that quantify
the degree of comonotonicity in a random vector.
Theorem 2.2 If ¹ is a permutation invariant ¯nite Borel measure on [0;1]n such that
0 <
R
(Cc ¡ Ci)d¹ < 1 then
m(C) =
R
(C ¡ Ci)d¹ R
(Cc ¡ Ci)d¹
is a measure of positive dependence de¯ned on C+(n).
Proof
Thanks to the previous observations, m(C)=0 if and only if C = Ci and m(C)=1 if and
only C = Cc. The monotonicity property follows easily from the de¯nition of the measure
m. Moreover we note that
R
(Cµ ¡ Ci)d¹ =
R
(C ¡ Ci)d(¹ ± fµ) =
R
(C ¡ Ci)d¹ and then
m(Cµ) = m(C) and the measure is permutation invariant. Since every sequence of copulas
converging to a copula pointwise, does so uniformly (see [6]) then the continuity property
follows.
Finally we characterize measures of multivariate dependence that preserve convex sums of
copulas (see [3] for the case of bivariate measure of concordance).
Theorem 2.3 A measure of positive dependence m de¯ned on C+(n) is such that
m(tA + (1 ¡ t)B) = tm(A) + (1 ¡ t)m(B) for t 2 (0;1) and A;B 2 C+(n)
if and only if there exists a permutation invariant ¯nite Borel measure ¹ on [0;1]n such that
0 <
R
(Cc ¡ Ci)d¹ < 1 and
m(C) =
R
(C ¡ Ci)d¹ R
(Cc ¡ Ci)d¹
:
Proof
It is easy to prove that a function de¯ned on C+(n) by
m(C) =
R
(C ¡ Ci)d¹ R
(Cc ¡ Ci)d¹
where ¹ is a permutation invariant ¯nite Borel measure on [0;1]n such that 0 <
R
(Cc ¡
Ci)d¹ < 1 is a multivariate measure of positive dependence that preserves the convex sum
of copulas.
Now we suppose that m is a measure of positive dependence de¯ned on C+(n) such that
m(tA + (1 ¡ t)B) = tm(A) + (1 ¡ t)m(B) for t 2 (0;1) and A;B 2 C+(n)
8and m(Cc) = 1. Letting C = fC ¡ Ci : C 2 C+(n)g and ¡ the functional de¯ned by
¡(C ¡ Ci) = m(C). The functional ¡ is bounded and linear and de¯ned on a compact and
convex subset of the space C([0;1]n) of continuous functions on [0;1]n with respect to k¢k1
and then it can be extended to a bounded linear functional on C([0;1]n). By the Riesz
Representation Theorem, there exists a Borel measure ¹ on [0;1]n such that
¡(A) =
Z
Ad¹
for every A 2 C([0;1]n). Then m(C) =
R
(C ¡ Ci)d¹ for any C 2 C+(n).
Moreover as in the proof of Theorem 3.3 we can prove that ¹ is permutation invariant.
References
[1] A. Colangelo, M. Scarsini and M.Shaked Some notions of multivariate positive depen-
dence Insurance: Mathematics and E conomics, 37, 2005, 13 - 26.
[2] H. Edwards, P. Mikusinski and M.D. Taylor, Measures of concordance determined by
D4 - invariant copulas, Proceedings of the American Mathematical Society, 33, 2005,
1505 - 1513.
[3] H. Edwards, P. Mikusinski and M.D. Taylor, Measures of concordance determined by
D4 - invariant measures on (0;1)2, Journal of Mathematics and Mathematical Sciences,
70, 2004, 3867 - 3875.
[4] H. Joe, Multivariate concordance, Journal of Multivariate Analysis, 35, 1990, 12 - 30.
[5] H. Joe, Multivariate Models and Dependence Concepts, Chapman & Hall, London,
1997.
[6] R B. Nelsen, An Introduction to Copulas, in: Lecture Notes in Statistics, Vol. 139,
Springer, New York 1999.
[7] I. Koch and A. De Schepper, The comonotonicity coe±cent: a new measure of positive
dependence in a multivariate setting, Research Paper Universiteit Antwerpen, Faculty
of Applied Economics 30, 2006.
[8] M. Scarsini, On measures of concordance, Stochastica, 8, 1984, 201 - 218.
[9] M.D. Taylor, Multivariate measures of concordance, Annals of the Institute of Statis-
tical Mathematics, 59, 2007, 789 - 806.
9

An Introduction to Copulas, in:

Measures of concordance determined by D4 - invariant copulas,

Measures of concordance determined by D4 - invariant measures on (0;1)2,

Multivariate concordance,

Multivariate measures of concordance,

Multivariate Models and Dependence Concepts,

On measures of concordance,

Scarsini and M.Shaked Some notions of multivariate positive dependence Insurance:

The comonotonicity coe±cent: a new measure of positive dependence in a multivariate setting, Research Paper Universiteit Antwerpen,

Multivariate measures of positive dependence

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