copulas and quasi-copulas with given diagonal sections Insurance: Mathematics and Economics 42 (2008) 473483
www.elsevier.com/locate/ime
On the construction of copulas and quasi-copulas with given diagonal
sections
$
Roger B. Nelsen
a
, Jos�e Juan Quesada-Molina
b
, , Jos�e Antonio Rodr�guez-Lallena
c
,
Manuel �
Ubeda-Flores
c
a
Department of Mathematical Sciences, Lewis & Clark College, 0615 S.W. Palatine Hill Road, Portland, OR 97219, USA
b
Departamento de Matem�atica Aplicada, Universidad de Granada, 18071 Granada, Spain
c
Departamento de Estad�stica y Matem�atica Aplicada, Universidad de Almer�a, 04120 Almer�a, Spain
Received September 2006; received in revised form November 2006; accepted 26 November 2006
Abstract
We study a method, which we call a copula (or quasi-copula) diagonal splice, for creating new functions by joining portions of two copulas
(or quasi-copulas) with a common diagonal section. The diagonal splice of two quasi-copulas is always a quasi-copula, and we nd a necessary
and sufcient condition for the diagonal splice of two copulas to be a copula. Applications of this method include the construction of absolutely
continuous asymmetric copulas with a prescribed diagonal section, and determining the best-possible upper bound on the set of copulas with a
particular type of diagonal section. Several examples illustrate our results.
c 2006 Elsevier B.V. All rights reserved.
MSC:
60E15; 62E10; 62H20
Keywords:
Bounds; Copulas; Diagonal sections; Distribution functions; Quasi-copulas
1. Introduction
The construction of distributions with given marginals has
been a problem of interest to statisticians for many years.
Today, in view of Sklars theorem (
Sklar, 1959
), this problem
can be reduced to the construction of a copula.
Nelsen
(
2006
)
summarizes different methods of constructing copulas. Copulas
have been used, among many other purposes, to nd best-
possible bounds on sets of distribution functions: see, for
instance,
Nelsen et al.
(
2004
),
Nelsen and �
Ubeda-Flores
(
2005
)
and
Rodr�guez-Lallena and �
Ubeda-Flores
(
2004
). In this paper
we will only deal with bivariate copulas and quasi-copulas.
Thus, in the sequel we will usually omit the word bivariate.
$
This work was partially supported by the Ministerio de Ciencia y
Tecnolog�a (Spain) and FEDER, under research project BFM200306522, and
also by the Consejer�a de Educaci�on y Ciencia of the Junta de Andaluc�a
(Spain). Corresponding author. Tel.: +34 958 249022; fax: +34 958 249513.
E-mail addresses:
nelsen@lclark.edu
(R.B. Nelsen),
jquesada@ugr.es
(J.J. Quesada-Molina),
jarodrig@ual.es
(J.A. Rodr�guez-Lallena),
mubeda@ual.es
(M. �
Ubeda-Flores).
A copula is a function C: [0
, 1]
2
[
0
, 1] which satises:
(C1) the boundary conditions C
(t, 0) = C(0, t) = 0 and
C
(t, 1) = C(1, t) = t for all t in [0, 1]; and
(C2) the 2-increasing property, i.e., V
C
([u
1
, u
2
] � [
v
1
, v
2
]
) =
C
(u
2
, v
2
) C(u
2
, v
1
) C(u
1
, v
2
) + C(u
1
, v
1
) 0 for
all u
1
, u
2
, v
1
, v
2
in [0
, 1] such that u
1 u
2
and
v
1 v
2
.
The rectangle [u
1
, u
2
] � [
v
1
, v
2
]
is called a 2-box. Equivalently,
a copula is the restriction to [0
, 1]
2
of a continuous bivariate
distribution function whose margins are uniform on [0
, 1].
The importance of copulas as a tool for statistical analysis
and modelling stems largely from the observation that
the joint distribution H of the random pair
(X, Y ) with
respective margins F and G can be expressed by H
(x, y) =
C
(F(x), G(y)) for every (x, y) [, ]
2
, where C is a
copula that is uniquely determined on Range F � Range G
(Sklars theorem).
If the random variables are exchangeable, i.e., if the random
vectors
(X, Y ) and (Y, X) are identically distributed, then the
copula C of
(X, Y ) is symmetric, i.e., C(u, v) = C(v, u)
for all
(u, v) [0, 1]
2
. Observe that, given a copula C, the
0167-6687/$ - see front matter c 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.insmatheco.2006.11.011 474
R.B. Nelsen et al. / Insurance: Mathematics and Economics 42 (2008) 473483
function C
t
dened on [0
, 1]
2
by C
t
(u, v) = C(v, u) is also
a copula. C
t
will be called the transpose of the copula C.
Note that a copula C is symmetric if and only if C
t
=
C
.
Many, perhaps most, of the copulas encountered in the literature
are symmetric. However, exchangeability of random variables
is rather uncommon in real life. In this paper we deal with
the common situation in which the copula C is asymmetric,
i.e., C
t
=
C
. The denitions of symmetric and asymmetric
copulas, and the transpose of a copula can be extended to any
function dened on [0
, 1]
2
.
The concept of a quasi-copula is a more general notion than
that of a copula. It was introduced by
Alsina et al.
(
1993
)
see
Nelsen et al.
(
1996
) for the multivariate case in order
to characterize operations on distribution functions that can or
cannot be derived from operations on random variables dened
on the same probability space. A quasi-copula see
Genest
et al.
(
1999
) for more details is a function Q: [0
, 1]
2
[
0
, 1]
which satises condition (C1), but in place of (C2), the weaker
conditions:
(Q1) Q is nondecreasing in each variable; and
(Q2) the Lipschitz condition
|
Q
(u
1
, v
1
) Q(u
2
, v
2
)| |u
1 u
2
| + |
v
1 v
2
|
for all
(u
1
, v
1
), (u
2
, v
2
) in [0, 1]
2
.
While every copula is a quasi-copula, there exist proper quasi-
copulas, i.e., quasi-copulas which are not copulas.
In the sequel, for any two functions A and B dened on a
common domain D, A B will denote A
(x) B(x) for every
x D.
Let W , and M denote the copulas dened by W
(u, v) =
max
(u + v 1, 0), (u, v) = uv and M(u, v) = min(u, v)
for every
(u, v) [0, 1]
2
. W and M are known as the
Fr�echetHoeffding bounds
for copulas and quasi-copulas, since
W Q M
for any quasi-copula (in particular, for any
copula) Q; and is known as the independence copula:
see
Nelsen
(
2006
) for more details.
The diagonal section C
of a copula C (and similarly for
a quasi-copula) is the function dened by C
(t) = C(t, t) for
every t [0
, 1]. On the other hand, a diagonal is a function
: [0, 1] R which satises the following conditions:
(i)
(1) = 1,
(ii)
(t) t for every t [0, 1], and
(iii) 0
(t ) (t) 2(t t) for every t, t [0, 1] such
that t t .
The diagonal section of any quasi-copula (or copula) is a
diagonal; and for any diagonal
, there exist copulas (and quasi-
copulas) whose diagonal section is
(
Fredricks and Nelsen,
1997
;
Nelsen and Fredricks, 1997
;
Nelsen et al.
,
2004
).
The diagonal section of a copula C has several probabilistic
interpretations (
Nelsen
,
2006
;
Nelsen et al.
,
2001
,
2004
); for
instance, C
is the restriction to [0
, 1] of the distribution
function of max
(U, V ) whenever (U, V ) is a random pair
distributed as C. More generally, if
(X, Y ) is distributed
according to H , with respective margins F and G and copula C,
and x
t
, y
t
R are respective 100t-th percentiles for every t
(0, 1), then
C
(t) = Pr[X x
t
, Y y
t
]
for every t
(0, 1).
Furthermore, C
can be used to study the tail dependence of
the random pair
(X, Y ) (
Nelsen, 2006
): the upper and lower
tail dependence parameters U
and L
, which are dened as U
=
lim
t
1 Pr[Y
> y
t
|
X
> x
t
]
and L
=
lim
t
0
+
Pr[Y
y
t
|
X x
t
]
(if the limits exist), can be computed as follows: U
=
2 C
(1 ) and
L
= C
(0
+
).
Copulas are used to build models for dependence between
risks in nancial and actuarial risk management, especially
dependence between extreme events (
B�auerle and M�uller, 1998
;
Denuit et al.
,
2005
;
Frees and Valdez, 1998
;
Klugman and
Parsa, 1999
). Tail dependence has been shown to be useful for
describing this dependence, in particular in volatile and bear
markets (
An�e and Kharoubi, 2003
;
Malevergne and Sornette,
2006
), and in contagion and stress testing concepts (
Abdous
et al., 2005
). For other applications, see
Embrechts et al.
(
2002
),
Frahm et al.
(
2005
),
Frahm et al.
(
2003
) and
Schmidt
(
2002
). Since, as we have just observed, tail dependence is a
property of the diagonal section of the copula in the model,
creating copulas with given diagonal section but with a variety
of dependence structures has applications in insurance and
nance.
In Section
2
, after some preliminary concepts and results, we
introduce a method for constructing copulas and quasi-copulas
with a given diagonal section. An alternative approach to that
method can be found in an unpublished paper of
Durante et al.
(
2007
). We show that, in particular, such a method can be used
to construct absolutely continuous asymmetric copulas.
In Section
3
, we rst review best-possible bounds for the sets
of all copulas or quasi-copulas with a common diagonal section
(see
�
Ubeda-Flores
(
2001
) for a preliminary study). In this study,
the only problematic case is the obtaining of the best-possible
upper bound for the set of copulas with a given diagonal section.
We introduce the concept of a simple diagonal and show that
many of the most commonly used copulas have simple diagonal
sections. We also show that an important subclass of such
diagonals is the convex diagonals. We nd an elementary way
to construct asymmetric copulas with simple diagonal sections
and, as an application, we obtain the best-possible upper bound
for the set of copulas with a given simple diagonal.
2. Construction of copulas with a given diagonal section
We begin this section with some notation which will be
useful in the sequel.
Consider the triangles T
U
and T
L
in [0
, 1]
2
, dened by T
U
=
{
(u, v) [0, 1]
2
:
u
v} and T
L
= {
(u, v) [0, 1]
2
:
u
v};
and their intersection D = T
L T
U
, which is the diagonal
of the unit square given by D = {
(u, u) [0, 1]
2
:
u
[
0
, 1]}. For any u, v [0, 1], [{u, v}] d