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Statistics & Probability Letters 66 (2004) 315–325
A new class of bivariate copulas?
Jos? e Antonio Rodr? ?guez-Lallena, Manuel?Ubeda-Flores∗
Departamento de Estad? ?stica y Matem? atica Aplicada, Universidad de Almer? ?a, Carretera de Sacramento s/n 04120,
La Ca˜ nada de San Urbano, Almer? ?a, Spain
Received February 2003; accepted September 2003
Abstract
We study a wide class of bivariate copulas depending on two univariate functions which generalizes many
known families of copulas. We measure the dependence of any copula of this class in di?erent ways, exhibit
several properties concerning symmetry, dependence concepts, and concordance ordering, and provide several
examples.
c ? 2003 Elsevier B.V. All rights reserved.
MSC: primary 62H05; secondary 62H20
Keywords: Concordance ordering; Copulas; Dependence concepts; Measures of association; Symmetry properties
1. Introduction
The purpose of this paper is to study a class of, in general, asymmetric bivariate copulas, which
generalizes several families such as the known Farlie–Gumbel–Morgenstern family of copulas and
others (e.g. Quesada-Molina and Rodr? ?guez-Lallena, 1995; Rodr? ?guez-Lallena, 1996; Nelsen et al.,
1997; Lai and Xie, 2000; Amblard and Girard, 2001, 2002). This class of copulas is interesting since
it provides an easy manner to construct bivariate models with a variety of dependence structures. A
preliminary study can be found in?Ubeda-Flores (1998).
A (bivariate) copula is a function C :I2→ I(=[0;1]) which satis?es the boundary conditions
C(t;0) = C(0;t) = 0andC(t;1) = C(1;t) = t;t ∈I
(1)
?Research supported by a Spanish C.I.C.Y.T. Grant (PB 98-1010) and the Consejer? ?a de Educaci? on y Ciencia of the
Junta de Andaluc? ?a (Spain).
∗Corresponding author.
E-mail address: mubeda@ual.es (M.?Ubeda-Flores).
0167-7152/$-see front matter c ? 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2003.09.010
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316 J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325
and the 2-increasing property, i.e.,
C(u2;v2) − C(u2;v1) − C(u1;v2) + C(u1;v1)¿0
for all u1;u2;v1;v2 in I such that u16u2 and v16v2. Equivalently, a copula is the restriction to
I2of a continuous bivariate distribution function whose margins are uniform on I. A copula C is
symmetric if C(u;v) = C(v;u) for all (u;v) in I2, whereas C is asymmetric otherwise.
The importance of copulas is described in Sklar’s theorem (Sklar, 1959): Let X and Y be random
variables with joint distribution function H and marginal distribution functions F and G, respectively.
Then there exists a copula C (which is uniquely determined on Range F × RangeG) such that
H(x;y) = C(F(x);G(y)) for all x;y in [ − ∞;∞]. Thus copulas link joint distribution functions
to their margins. Copulas are of interest to statisticians namely for two reasons: First, copulas are
a way of studying scale-free measures of dependence, and secondly, copulas are a tool to build
families of bivariate distributions with given margins (Fisher, 1997). For further details about copulas,
see Nelsen (1999).
Let X and Y be continuous random variables with joint distribution function H, margins F and
G, respectively, and associated copula C (via Sklar’s theorem). It is known that X and Y are
exchangeable if and only if F = G and C is symmetric. Thus, the class that we study in this paper
is mainly made up of copulas which insure the non-exchangeability of two identically distributed
continuous random variables; but it also contains a variety of families of symmetric copulas, as we
show in Section 4.
Finally, we introduce some notation. Let ? denote the copula of independent random variables,
i.e., ?(u;v) = uv for all (u;v) in I2. Let n¿1 and let P and Q be two functions with common
domain A ⊂ Rn. Then P 6Q (or Q¿P) means that P(x)6Q(x) for all x∈A; in particular, P 60
stands for P(x)60 for all x∈A.
In Section 2 we describe a new class of copulas. In Section 3, we study the concordance ordering
and several dependence concepts, symmetry properties and measures associated with copulas in this
class. Finally, in Section 4, we provide several examples.
(2)
2. A new class of copulas
Let f and g be two real functions de?ned on I and consider the function C given by
C(u;v) = uv + f(u)g(v) (3)
for all u;v in I. Our purpose is to determine the cases in which C is a copula. For that we need two
preliminary lemmas. The ?rst one is a well-known result from real analysis (Stromberg, 1981).
Lemma 2.1. Let f:[a;b] → R and ?;?∈R. Then the following statements are equivalent:
(1) ?(y − x)6f(y) − f(x)6?(y − x) for all x;y∈[a;b] such that x¡y,
(2) f is absolutely continuous on [a;b] and ?6f?(x)6? for almost every x∈[a;b].
The following result is a consequence of Lemma 2.1.
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J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325317
Lemma 2.2. Let f:[a;b] → R be absolutely continuous and let A = {x∈[a;b]: f?(x) exists}.
Then:
(1) inf{(f(y) − f(x))=(y − x): x;y∈[a;b];x¡y} = inf{f?(x): x∈A},
(2) sup{(f(y) − f(x))=(y − x): x;y∈[a;b];x¡y} = sup{f?(x): x∈A}.
We are now in position to state and prove the main result of this section.
Theorem 2.3. Let f and g be two non-zero real functions de?ned on I. Let C :I2→ R be the
function de?ned by (3). Then C is a copula, if and only if
(1) f(0) = f(1) = g(0) = g(1) = 0,
(2) f and g are absolutely continuous and
(3) min{??;??}¿−1, where ? =inf{f?(u): u∈A}¡0, ? =sup{f?(u): u∈A}¿0, ?=inf{g?(v):
v∈B}¡0 and ? = sup{g?(v): v∈B}¿0, with A = {u∈I: f?(u) exists} and B = {v∈I: g?(v)
exists}. Furthermore, in such a case, C is absolutely continuous.
Proof. It is immediate that the function given by (3) satis?es the boundary conditions (1) for copulas
if and only if f(0)=f(1)=g(0)=g(1)=0. We now prove that C is 2-increasing if and only if both
statements 2 and 3 hold. Let F and G denote the functions de?ned on the set T ={(x;y)∈I2: x¡y}
by F(x;y) = (f(y) − f(x))=(y − x) and G(x;y) = (g(y) − g(x))=(y − x). Let (u1;u2);(v1;v2) be in
T. Then, C satis?es condition (2), if and only if
− 16F(u1;u2)G(v1;v2):
Hence, C is 2-increasing if and only if the following two inequalities hold:
(4)
− 16sup{F(u1;u2): u1¡u2;f(u1)¡f(u2)}inf{G(v1;v2): v1¡v2;g(v1)¿g(v2)};
− 16inf{F(u1;u2): u1¡u2;f(u1)¿f(u2)}sup{G(v1;v2): v1¡v2;g(v1)¡g(v2)}:
The sets in (5) and (6) are non-empty since f(0)=f(1)=g(0)=g(1)=0 and f and g are non-zero.
On the other hand, since (4) is satis?ed for all (v1;v2) in T, we have
(5)
(6)
sup{−1=G(v1;v2): v1¡v2;g(v1)¡g(v2)}6F(u1;u2)
6inf{−1=G(v1;v2): v1¡v2;g(v1)¿g(v2)}
for all (u1;u2) in T. Similar bounds can be obtained for G(v1;v2). Therefore, the suprema and
in?ma in (5) and (6) are ?nite. Furthermore, from Lemma 2.1 we have that f and g are absolutely
continuous; and from Lemma 2.2 we have
sup{F(u1;u2): u1¡u2;f(u1)¡f(u2)}=sup{F(u1;u2): u1¡u2}
=sup{f?(u): u∈A} = ?¿0;
inf{F(u1;u2): u1¡u2;f(u1)¿f(u2)}=inf{F(u1;u2): u1¡u2}
=inf{f?(u): u∈A} = ?¡0;
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318J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325
sup{G(v1;v2): v1¡v2;g(v1)¡g(v2)}=sup{G(v1;v2): v1¡v2}
=sup{g?(v): v∈B} = ?¿0;
inf{G(v1;v2): v1¡v2;g(v1)¿g(v2)}=inf{G(v1;v2): v1¡v2}
=inf{g?(v): v∈B} = ?¡0:
In short, we have obtained that, if C is a copula, then f and g are absolutely continuous and
min{??;??}¿−1. Conversely, we only need to follow the same steps backwards, which completes
the proof.
In the following, the set of copulas of the class characterized in Theorem 2.3 will be denoted
by C. Several known parametric families of copulas are included in the set C, as we said in the
Introduction. The following corollary, whose proof is straightforward, provides many new parametric
families in C.
Corollary 2.4. Let f and g be two non-zero absolutely continuous functions de?ned on I such that
f(0) = f(1) = g(0) = g(1) = 0. Let C?be the function de?ned on I2by C?(u;v) = uv + ?f(u)g(v),
with ?∈R. Then, C?is a copula if and only if −1=max{??;??}6?6 − 1=min{??;??}.
The next result provides su?cient conditions on the functions f and g in order that the function
C de?ned by (3) is a copula.
Theorem 2.5. Let f and g be two non-zero real functions de?ned on I such that f(0) = f(1) =
g(0) = g(1) = 0 and satisfying the Lipschitz conditions
|g(u1) − g(u2)|6|u1− u2|
for all u1;u2in I with M ¿0. Then the function C given by (3) is an absolutely continuous copula.
|f(u1) − f(u2)|6M|u1− u2|
and
M
(7)
Proof. The function C satis?es the boundary conditions (1) from the fact that f(0)=f(1)=g(0)=
g(1)=0. Let F;G and T be as in the proof of Theorem 2.3. Let (u1;u2);(v1;v2) be in T. From the
Lipschitz conditions (7) we have 1¿|F(u1;u2)||G(v1;v2)|¿−F(u1;u2)G(v1;v2), i.e., the functions
F and G satisfy condition (4), whence C is 2-increasing, as required.
However, the converse of Theorem 2.5 does not hold in general, as the following example
shows:
Example 2.1. Let m and n be two real numbers such that mn¿0 with mn ?= 1, and let f and g be
the functions de?ned by
?mu;
(1 − u)=n; 1=(mn + 1)6u61;
f(u) =
06u¡1=(mn + 1);
and g(v) =
?nv;
(1 − v)=m; 1=(mn + 1)6v61;
06v¡1=(mn + 1);
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J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325319
respectively. Both f and g are continuous and either concave or convex, whence they are absolutely
continuous (Stromberg, 1981). In this case, the values of ?, ?, ? and ? de?ned in Theorem 2.3
are the following: ?=min{m;−1=n}, ? =max{m;−1=n}, ?=min{n;−1=m} and ?=max{n;−1=m},
whence min{??;??} = −1. Therefore, the function C de?ned by (3), i.e.,
is a copula. However, if mn¿1 and we take u1;u2in I such that u1?= u2and max(u1;u2)¡1=(mn+
1), then we have |f(u1) − f(u2)||g(u1) − g(u2)| = mn|u1− u2|2¿|u1− u2|2, which contradicts (7);
similarly, assuming mn¡1 and taking u1;u2 in I such that u1?= u2 and min(u1;u2)¿1=(mn + 1),
then |f(u1) − f(u2)||g(u1) − g(u2)| = |u2− u1|2=(mn)¿|u2− u1|2.
Example 2.1 also shows that the characterization for copulas of form (3) postulated by Amblard
and Girard (2002) is not correct.
As a consequence of Theorem 2.5 we have the following:
C(u;v) =
uv(mn + 1);06u;v¡1=(mn + 1);
uv + (1 − u)(1 − v)=(mn);
min(u;v);
1=(mn + 1)¡u;v61;
otherwise
(8)
Corollary 2.6. Let M ¿0, and let f and g be two non-zero real functions de?ned on I such that
(1) f and g are absolutely continuous,
(2) |f?(u)|6M and |g?(u)|61=M a.e. in I,
(3) |f(u)|6M min{u;1 − u} and |g(u)|6(1=M)min{u;1 − u} for all u in I.
Then the function C de?ned by (3) is an absolutely continuous copula.
3. Properties of the new class of copulas
We now study several properties of the family of the set C. They concern joint product moments,
concordance ordering, dependence concepts, symmetry and measures of association.
Let C be a copula in the set C. Then, the density of C is given by c(u;v) = 1 + f?(u)g?(v), a.e.
in I2. The next result follows immediately from this fact.
Theorem 3.1. Let (U;V) be a pair of uniform I random variables with associated copula C ∈C.
Then
1
(i + 1)(j + 1)+ ij
0
and, as a consequence, the covariance and the correlation coe?cient of the pair (U;V) are
given by
?1
respectively.
E(UiVj) =
?1
ui−1f(u)du
?1
0
vj−1g(v)dv
Cov(U;V) =
0
f(u)du
?1
0
g(v)dv andCorr(U;V) = 12
?1
0
f(u)du
?1
0
g(v)dv;
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320J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325
The population versions of four of the most common non-parametric measures of association
between the components of a continuous random pair (X;Y) are Kendall’s tau, Spearman’s rho,
Gini’s gamma and the Spearman’s footrule coe?cient. Such measures depend only on the copula C
of the pair (X;Y), and are given by ?C=4?1
The following theorem, whose proof is simple, provides expressions for those measures when we
consider a copula in C.
0
?1
0C(u;v)dC(u;v)−1, ?C=12?1
0
?1
0C(u;v)dudv−3,
?C= 4?1
0[C(u;u) + C(u;1 − u)]du − 2, and ’C= 6?1
0C(u;u)du − 2, respectively (Nelsen, 1999).
Theorem 3.2. Let C be a copula in the set C. Then the values of Kendall’s tau, Spearman’s
rho, Gini’s gamma and Spearman’s footrule associated with C are respectively given by ?C=
8?1
0f(t)dt?1
0g(t)dt, ?C= 12?1
0f(t)dt?1
0g(t)dt = 3?C=2, ?C= 4?1
0f(t)[g(t) + g(1 − t)]dt and
’C= 6?1
Example 3.1. Let m and n be two real numbers such that mn¿0.
0f(t)g(t)dt.
(a) If C is the copula given by (8), then Theorem 3.2 yields ?C= ’C= 2mn=(mn + 1)2, ?C=
3mn=(mn+1)2and ?C=2max{mn(3−mn);3−1=mn}=(3mn+3). It is clear that such measures
reach all the values in the intervals (0;1
are evaluated for mn∈(0;1].
(b) Let f be the function given in Example 2.1 and let g be the function de?ned by
2], (0;3
4] and (0;2
3], respectively, when, for instance, they
g(v) =
?−v=m;
n(v − 1);
06v6mn=(mn + 1);
mn=(mn + 1)¡v61:
These functions satisfy the hypotheses of Theorem 2.3, whence the function C given by (3) is
a copula. For this copula, Theorem 3.2 yields ?C= −2mn=(mn + 1)2, ?C= −3mn=(mn + 1)2,
?C=2min{mn(mn−3);(1=mn)−3}=(3mn+3) and ’C=min{mn(−1−2mn+(mn)2);(1−2mn−
(mn)2)=mn}=(mn + 1)2. These measures reach all the values in the intervals [ −1
[ −2
From Example 3.1, the range of the measures ?C, ?C, ?Cand ’C, with C ∈C, contains respectively
the intervals [−1
such intervals.
We now investigate a partial ordering on C. Let C1 and C2 be two copulas. C2 is said more
concordant than C1, written C1≺ C2, if C16C2 (Joe, 1997). Thus, if C1 and C2 are two copulas
in C, say C1(u;v)=uv+f1(u)g1(v) and C2(u;v)=uv+f2(u)g2(v) for all (u;v) in I2, then C1≺ C2
if and only if f1(u)g1(v)6f2(u)g2(v) for all (u;v) in I2. The next result provides simple su?cient
conditions on the functions f1;g1;f2and g2for C1and C2to be ordered by ≺.
Corollary 3.3. Let C1and C2be two copulas in C such that C1(u;v)=uv+f1(u)g1(v) and C2(u;v)=
uv + f2(u)g2(v) for all (u;v) in I2. If, for some ? ?= 0, it is satis?ed that 06f16?f2 and
06g16(1=?)g2, or ?f26f160 and (1=?)g26g160, then C1≺ C2.
2;0), [ −3
4;0),
3;0) and [ −1
2;0), respectively, when, for instance, they are evaluated for mn∈(0;1].
2;1
2], [−3
4;3
4], [−2
3;2
3] and [−1
2;1
2]. We conjecture that those ranges coincide with
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J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325 321
A parametric family of copulas {C?} is positively (negatively) ordered when ?16?2implies that
C?1≺ C?2(C?2≺ C?1): see Nelsen (1999). For the families of copulas given by Corollary 2.4, we
have the following result, whose proof is straightforward.
Theorem 3.4. Let f and g be two non-zero absolutely continuous functions de?ned on I such that
f(0)=f(1)=g(0)=g(1)=0. Let {C?} be the corresponding family of copulas given by Corollary
2.4. Then, {C?} is positively (negatively) ordered, if and only if either f¿0 and g¿0, or f60
and g60 (either f¿0 and g60, or f60 and g¿0).
The following two theorems consider continuous random pairs with associated copula C in C, and
characterize those pairs which satisfy certain well-known positive dependence properties (it can be
obtained similar results for the corresponding negative dependence concepts). For their de?nitions,
see Joe (1997) and Nelsen (1999). The proofs of both theorems rely on the expression of those
properties in terms of the copula associated with the random pair (Nelsen, 1999). Thus, the proof
of the ?rst result is straightforward.
Theorem 3.5. Let (X;Y) be a continuous random pair whose associated copula C belongs to C.
Then X and Y are positively quadrant dependent (PQD(X;Y)), i.e., ? ≺ C, if and only if either
f¿0 and g¿0, or f60 and g60.
Every positive dependence concept that we study in the following theorem implies PQD(X;Y):
see Nelsen (1999). Thus, we can suppose without loss of generality that the functions f and g are
both non-negative.
Theorem 3.6. Let (X;Y) be a continuous random pair whose associated copula C is in C, and such
that both the functions f and g are non-negative. Then:
(1) Y is left tail decreasing in X (LTD(Y |X)) if and only if f(u)¿uf?(u) a.e. in I, and
LTD(X |Y) holds if and only if g(v)¿vg?(v) a.e. in I.
(2) Y is right tail increasing in X (RTI(Y |X)) if and only if f(u)¿(u − 1)f?(u) a.e. in I, and
RTI(X |Y) holds if and only if g(v)¿(v − 1)g?(v) a.e. in I.
(3) Y is stochastically increasing in X (SI(Y |X)) if and only if f is concave, and SI(X |Y) holds
if and only if g is concave.
(4) X and Y are left corner set decreasing (LCSD(X;Y)) if and only if both LTD(Y |X) and
LTD(X |Y) hold.
(5) X and Y are right corner set increasing (RCSI(X;Y)) if and only if both RTI(Y |X) and
RTI(X |Y) hold.
(6) X and Y are positively likelihood ratio dependent (PLR(X;Y)) if and only if both SI(Y |X)
and SI(X |Y) hold.
Proof. The proof of the ?rst three statements is immediate. Let us prove statement 4. From Corollary
5.2.17 in Nelsen (1999), LCSD(X;Y) holds, if and only if
[u1f(u2) − u2f(u1)][v1g(v2) − v2g(v1)]¿0(9)
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322J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325
for all u1;u2;v1;v2 in I such that u1¡u2 and v1¡v2. First, suppose that both LTD(Y |X) and
LTD(X |Y) hold. Then, the functions f(u)=u and g(v)=v are non-increasing on I, whence u1f(u2)−
u2f(u1)60 and v1g(v2) − v2g(v1)60 for all u1;u2;v1;v2 in I such that u1¡u2 and v1¡v2, i.e.,
the inequality (9) holds. Conversely, the result is trivial since LCSD(X;Y) always implies both
LTD(Y |X) and LTD(X |Y) (Nelsen, 1999, Theorem 5.2.14). Statements 5 and 6 can be proved
using similar arguments.
The next result studies properties of symmetry in any continuous random pair (X;Y) whose
associated copula C belongs to the set C. For a complete review of those symmetry properties
and their characterizations in terms of the copula associated to the random pair, see Nelsen (1999).
Theorem 3.7. Let (a;b) be a point in R2and let (X;Y) be a pair of continuous random variables
whose associated copula C is given by (3), i.e., C ∈C.
(1) X and Y are exchangeable if and only if X and Y are identically distributed and there exists
? ?= 0 such that f(t) = ?g(t) for all t in I.
(2) Suppose that (X;Y) is marginally symmetric about (a;b). Then:
(a) (X;Y) is radially symmetric about (a;b) if and only if either f(t) = f(1 − t) and g(t) =
g(1 − t) for all t in I, or f(t) = −f(1 − t) and g(t) = −g(1 − t) for all t in I.
(b) (X;Y) is jointly symmetric about (a;b) if and only if f(t)=−f(1−t) and g(t)=−g(1−t)
for all t in I.
Proof. Let us prove statement 2(a) (the proof of statement 1 is similar and 2(b) is immediate).
From Theorem 2.7.3 in Nelsen (1999), the pair (X;Y) is marginally symmetric about (a;b) if and
only if f(u)g(v)=f(1−u)g(1−v) for all (u;v) in I2. Let u0be a point in I such that f(u0) ?= 0.
We denote ?=f(1−u0)=f(u0). Then we have g(v)=?g(1−v) for all v in I (observe that ? ?= 0; if
this were not the case, we would have that g(v)=0 for all v in I, which contradicts the hypothesis
that g is a non-zero function). In particular, g(v) = ?g(1 − v) = ?2g(v) for all v in I, whence ? = 1
or −1. A similar argument yields that f(u)=f(1−u) or f(u)=−f(1−u) for all u in I, whence
the proof follows.
4. Additional examples
We begin this section studying a new parametric family of copulas in C which covers various types
of dependence and symmetry structures, and such that it contains, as a particular case, the family
studied by Lai and Xie (2000). We also compute several measures of the dependence associated
with that parametric family.
Example 4.1. Let (X;Y) be a continuous random pair whose associated copula C? is given
by C?(u;v) = uv + ?uavb(1 − u)c(1 − v)dfor every (u;v) in I2, with a;b;c;d¿1. From Corollary
2.4 and after many calculations, it can be proved that C? is a copula if and only if
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−1=max{??;??}6?6−1=min{??;??}, where ?=−?=1 if a=c=1, ?=−?=1 if b=d=1 and
?=−
a + ca(a + c − 1)
?
?=
a + ca(a + c − 1)
?
?
?d−1 ?
?b−1?
?d−1 ?
otherwise. Moreover, the range for ? contains the interval [ − 1;1] for all a;b;c;d¿1. The case
a=b=c=d=1 produces the Farlie–Gumbel–Morgenstern family of copulas (and the smallest range
for ?, i.e., the interval [ − 1;1]). In general, the bigger the parameters a;b;c;d are, the bigger the
range for ? is (for instance: if a = b = c = d = 2, then ?∈[ − 27;27]; if a = 2; b = 3; c = 4; d = 5,
then ?∈[ − 840:445;939:403]; etc.).
Theorem 3.2 produces the following results: ?C?= 8?Beta(a + 1;c + 1)Beta(b + 1;d + 1), ?C?=
12?Beta(a+1;c+1)Beta(b+1;d+1), ?C?=4?[Beta(a+d+1;b+c+1)+Beta(a+b+1;c+d+1)],
and ’C?= 6?Beta(a + b + 1;c + d + 1).
From Theorem 3.4, we have that the family C? is positively ordered with respect to the concor-
dance ordering. Theorem 3.5 yields that PQD(X;Y) holds if and only ?¿0. If ?¿0 Theorem 3.6
yields: (a) LTD(Y |X) holds if and only if a = 1; (b) LTD(X |Y) holds if and only if b = 1; (c)
RTI(Y |X) holds if and only if c=1; (d) RTI(X |Y) holds if and only if d=1; (e) SI(Y |X) holds
if and only if a = c = 1; (f) SI(X |Y) holds if and only if b = d = 1; (g) LCSD(X;Y) holds if and
only if a = b = 1; (h) RCSI(X;Y) holds if and only if c = d = 1; (i) PLR(X;Y) holds if and only
if a = b = c = d = 1.
Finally, from Theorem 3.7 we have that (X;Y) is exchangeable if and only if X and Y are
identically distributed, a=b and c=d. Furthermore, if the pair (X;Y) is marginally symmetric about
a point in R2, then (X;Y) is radially symmetric if and only if a=c and b=d; but (X;Y) is jointly
symmetric only for the case ? = 0.
?
a
?a−1?
?
?a−1?
?
b
b + d
1 +
?
c
?a−1?
ac
a + c − 1;
?a−1?
ac
a + c − 1;
?b−1?
c
a + c
?c−1
×
?
1 −
a
a
c(a + c − 1)
1 −
?c−1?
c
?
c
a + c
?c−1
×
1 +
a
c(a + c − 1)
?b−1?
?c−1?
?=−
1 +
?
d
b(b + d − 1)
d
b + d
?d−1
×
?
1 −
?
b
d(b + d − 1)
bd
b + d − 1;
?b−1?
?=
?
b
b + d
1 −
?
d
b(b + d − 1)
d
b + d
?d−1
×
?
1 +
?
b
d(b + d − 1)
bd
b + d − 1;
Page 10
324J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325
Now, we consider the symmetric copulas in C. From statement 1 in Theorem 3.7 and
Corollary 2.4, those copulas are given by
C(u;v) = uv + ?f(u)f(v);(u;v)∈I2; (10)
where f is an absolutely continuous function de?ned on I such that f(0) = f(1) = 0, and ? is a
real number such that −1=max{?2;?2}6?6 − 1=(??) (? and ? are de?ned as in Theorem 2.3).
This type of copula was introduced by Rodr? ?guez-Lallena (1996). Recently, Amblard and Girard
(2001, 2002) studied a large subset of those copulas, speci?cally the ones with parameter ? varying
in the interval [−1;1] and f satisfying the Lipschitz condition |f(x)−f(y)|6|x −y|, (x;y)∈I2.
From our results, the families studied by Amblard and Girard can be generally extended to a larger
range of the parameter ?; even more, the function f can satisfy hypotheses more general than the
ones used by those authors. The following example shows a symmetric copula in C which does not
belong to the set of copulas studied by Amblard and Girard.
Example 4.2. Let C be the copula de?ned by (8). We know that C is a symmetric copula in C.
Suppose that C is a copula of the type described by Amblard and Girard (2001, 2002). Then, there
exist a number ? in [−1;1] and a function ’ satisfying ’(0)=’(1)=0 and the Lipschitz condition
|’(x) − ’(y)|6|x − y|, (x;y)∈I2, such that C(u;v) = uv + ?’(u)’(v) for all (u;v) in I2. Then,
?(mn + 1)u2;
u2+ (1 − u)2=(mn);
whence ?∈(0;1], and ’(u) = ±min{u?mn=?;(1 − u)=√mn?}. Since max{?(mn)=?;1=√mn?}¿1,
does not belong to the type studied by Amblard and Girard.
C(u;u) = u2+ ?’2(u) =
06u61=(mn + 1);
1=(mn + 1)6u61;
’ does not satisfy the required Lipschitz condition. Therefore, C is a symmetric copula in C which
Finally, as an application, we study the symmetric copulas with cubic sections (Nelsen et al.,
1997) which belong to the set C. As it is easy to check, such copulas are given by (10), where
f(t) = t(1 − t)(? − t) for all t in I and ?∈R. After some computations, it can be obtained that
C(u;v)=uv+?uv(1−u)(1−v)(?−u)(?−v), (u;v)∈I2, is a copula if and only if ?∈[−(??)2;−????],
where ??= 1=min(? − 1;−?) and
??=
−1=?;
3=(?2− ? + 1);
1=(? − 1);
?6 − 1;
−16?62;
?¿2:
Acknowledgements
The authors thank two anonymous referees for insightful comments on the previous versions of
the manuscript.
Page 11
J.A. Rodr? ?guez-Lallena, M. ? Ubeda-Flores/Statistics & Probability Letters 66 (2004) 315–325325
References
Amblard, C., Girard, S., 2001. Une famille semi-param? etrique de copules sym? etriques bivari? ees. C. R. Acad. Sci. Paris
S? erie I 333, 129–132.
Amblard, C., Girard, S., 2002. Symmetry and dependence properties within a semiparametric family of bivariate copulas.
J. Nonparametr. Statist. 14, 715–727.
Fisher, N.I., 1997. Copulas. In: Kotz, S., Read, C.B., Banks, D.L. (Eds.), Encyclopedia of Statistical Sciences, Update
Vol. 1. Wiley, New York, pp. 159–163.
Joe, H., 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, New York.
Lai, C.D., Xie, M., 2000. A new family of positive quadrant dependent bivariate distributions. Statist. Probab. Lett. 46,
359–364.
Nelsen, R.B., 1999. An Introduction to Copulas. Springer, New York.
Nelsen, R.B., Quesada Molina, J.J., Rodr? ?guez Lallena, J.A., 1997. Bivariate copulas with cubic sections. J. Nonparametr.
Statist. 7, 205–220.
Quesada Molina, J.J., Rodr? ?guez Lallena, J.A., 1995. Bivariate copulas with quadratic sections. J. Nonparametr. Statist. 5,
323–337.
Rodr? ?guez Lallena, J.A., 1996. Estudio de la compatibilidad y dise˜ no de nuevas familias en la teor? ?a de c? opulas.
Aplicaciones. Servicio de Publicaciones de la Universidad de Almer? ?a, Spain.
Sklar, A., 1959. Fonctions de r? epartition ? a n dimensions et leurs marges. Publ. Inst. Statist. Paris 8, 229–231.
Stromberg, K.R., 1981. Introduction to Classical Real Analysis. Wadsworth & Brooks/Cole, Paci?c Grove, CA.
?Ubeda Flores, M., 1998. Introducci? on a la teor? ?a de c? opulas. Aplicaciones. Trabajo de investigaci? on de Tercer Ciclo,
Universidad de Almer? ?a, Spain.
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