We present a new way of constructing bivariate copulas, by recalling and gluing two (or more) copulas. Examples illustrate how this construction can be applied to build complicated copulas from simple ones. --

Siburg, Karl Friedrich

Stoimenov, Pavel A.

English

Research Papers in Economics

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Siburg, Karl Friedrich; Stoimenov, Pavel A.
Working Paper
Gluing copulas
Technical Report // Sonderforschungsbereich 475, Komplexitätsreduktion in Multivariaten
Datenstrukturen, Universität Dortmund, No. 2007,31
Provided in cooperation with:
Technische Universität Dortmund
Suggested citation: Siburg, Karl Friedrich; Stoimenov, Pavel A. (2007) : Gluing copulas,
Technical Report // Sonderforschungsbereich 475, Komplexitätsreduktion in Multivariaten
Datenstrukturen, Universität Dortmund, No. 2007,31, http://hdl.handle.net/10419/25014Gluing copulas
Karl Friedrich Siburg
Fachbereich Mathematik
UniversitÄ at Dortmund
Vogelpothsweg 87
44227 Dortmund
Germany
Pavel A. Stoimenov
Fachbereich Statistik
UniversitÄ at Dortmund
Vogelpothsweg 78
44227 Dortmund
Germany
September 6, 2007
Abstract
We present a new way of constructing bivariate copulas, by rescal-
ing and gluing two (or more) copulas. Examples illustrate how this
construction can be applied to build complicated copulas from simple
ones.
1 Introduction
Let I = [0;1] be the closed unit interval and I2 = [0;1] £ [0;1] the
closed unit square. A (two-dimensional) copula is a function C : I2 !
I satisfying the following conditions:
1. C(x;0) = C(0;y) = 0 for all x;y 2 I
2. C(x;1) = x and C(1;y) = y for all x;y 2 I
3. C is 2-increasing, i.e., C(x2;y2)¡C(x2;y1)¡C(x1;y2)+C(x1;y1) ¸
0 for all rectangles [x1;x2] £ [y1;y2] ½ I2.
These conditions imply further key properties of copulas. In particu-
lar, a copula is Lipschitz continuous and increasing in each argument;
therefore, its partial derivatives exist a.e. on I2. There are three dis-
tinguished copulas, namely
C¡(x;y) = max(x + y ¡ 1;0)
C+(x;y) = min(x;y)
P(x;y) = xy:
1C+ and C¡ are called the Fr¶ echet-Hoe®ding upper and lower bound,
respectively, since for any copula C and any (x;y) 2 I2 we have the
estimates
C¡(x;y) · C(x;y) · C+(x;y): (1)
The true importance of copulas to probability theory stems from
the well known Sklar theorem [4, 3] which states that, when a joint
distribution function has continuous marginal distribution functions,
it can be decomposed into the margins and a unique copula.
Theorem 1.1 (Sklar's theorem). For every two-dimensional distribu-
tion function H with marginal distribution functions F;G there exists
a copula C such that
H(x;y) = C(F(x);G(y)):
Moreover, if F and G are continuous then C is unique.
Conversely, given any copula C and distribution functions F;G the
above equation de¯nes a two-dimensional distribution function H with
marginal distribution functions F;G.
In view of this theorem, a rich collection of copulas yields an
equally rich collection of bivariate distribution functions with arbi-
trary margins, which proves useful in modeling and simulation.
There are several ways of constructing copulas. Among them are
geometric methods (e.g., ordinal sums, shu²es of min, or copulas with
prescribed diagonal sections), algebraic methods (e.g., a copula trans-
formation), and methods based on generators, leading to the large
class of Archimedean copulas (including the well known Frank and
Gumbel families); for details we refer to [3]. For less known construc-
tions see, e.g., [1, 2].
In this paper, we present a new method for constructing copulas,
the so-called gluing construction. In its simplest form, it proceeds as
follows. Given two copulas C1;C2 and a number µ 2 (0;1), the graphs
of C1 and C2 are rescaled and pasted into the rectangle [0;µ] £ I and
[µ;1] £ I, respectively, i.e., they are glued together at the vertical
fµg £ I.
This simple gluing construction can be generalized in various di-
rections. First of all, it is possible to glue together not just two but
actually countably many copulas at once. Secondly, one can also glue
copulas vertically along a horizontal segment I £ fµg. Finally, com-
bining both leads to the gluing method in its most general form.
22 The gluing method for two copulas
Consider two copulas C1;C2. For any given parameter µ 2 (0;1),
consider the partition I = [0;µ] [ [µ;1] and set
(C1 t
X
C2)(x;y) =
(
µC1(x
µ;y) if 0 · x · µ
(1 ¡ µ)C2
¡x¡µ
1¡µ;y
¢
+ µy if µ · x · 1
(2)
Thus, C1 t
X
C2 is the result of gluing C1 and C2 horizontally along the
x-axis. We claim that it is indeed a copula.
Theorem 2.1. For any two copulas C1;C2 and any µ 2 (0;1), the
function C1 t
X
C2 is again a copula.
Proof. It follows immediately from Ci(0;y) = Ci(x;0) = 0 that
(C1 t
X
C2)(0;y) = (C1 t
X
C2)(x;0) = 0
for all x;y 2 I. Moreover, it is also clear from the construction that
(C1 t
X
C2)(1;y) = y and (C1 t
X
C2)(x;1) = x
for every x;y 2 I, because the corresponding properties hold for C1
and C2.
Hence, the only condition to be checked is that C1 t
X
C2 is 2-
increasing. Abbreviating C = C1 t
X
C2, we have to show that
C(x2;y2) ¡ C(x2;y1) ¡ C(x1;y2) + C(x1;y1) ¸ 0 (3)
for all rectangles R = [x1;x2] £ [y1;y2] ½ I2. For this, we distinguish
two cases.
For the ¯rst case, we assume that R ½ [0;µ] £ I or R ½ [µ;1] £ I,
i.e., µ = 2 (x1;x2). Then (3) requires that
µ ¢
h
C1(
x2
µ
;y2) ¡ C1(
x2
µ
;y1) ¡ C1(
x1
µ
;y2) + C1(
x1
µ
;y1)
i
¸ 0
respectively
(1 ¡ µ) ¢
h
C2(
x2 ¡ µ
1 ¡ µ
;y2) ¡ C2(
x2 ¡ µ
1 ¡ µ
;y1) ¡ C2(
x1 ¡ µ
1 ¡ µ
;y2)
+ C2(
x1 ¡ µ
1 ¡ µ
;y1)
i
+ µ ¢ (y2 ¡ y1 ¡ y2 + y1) ¸ 0
Since the last term in the second inequality adds up to zero, each in-
equality follows from the fact that C1, respectively C2, is 2-increasing.
3( ,y ) q 1
( ,y ) q 2
(x ,y ) 1 1
(x ,y ) 1 2
(x ,y ) 2 1
(x ,y ) 2 2
0 1
1
q
x
y
Figure 1: For the proof that C1 t
X
C2 is 2-increasing
In the second case, where µ 2 (x1;x2), we introduce the two auxil-
iary points (µ;y1) and (µ;y2) (see Figure 1) and observe that (3) would
follow from the two inequalities
C(x2;y2) ¡ C(x2;y1) ¡ C(µ;y2) + C(µ;y1) ¸ 0
C(µ;y2) ¡ C(µ;y1) ¡ C(x1;y2) + C(x1;y1) ¸ 0:
Each single inequality can be treated as in the ¯rst case above and,
as before, follows from the fact that C1 and C2 are 2-increasing.
Of course, the gluing construction can be also done with respect
to the second coordinate which leads to vertical gluing. Namely, given
two copulas C1;C2 and some µ 2 (0;1), we de¯ne
(C1 t
Y
C2)(x;y) =
(
µC1(x;
y
µ) if 0 · y · µ
(1 ¡ µ)C2
¡
x;
y¡µ
1¡µ
¢
+ µx if µ · y · 1
The same calculations as above show that C1 t
Y
C2 is again a copula.
3 Generalizations
The gluing method described above can be generalized to in¯nite
partitions. Let fJkgk2N be a countable set of closed intervals Jk =
[ak;bk] ½ I with pairwise disjoint interior such that [kJk = I. There
are two possible cases:
1. ak < bk for all k 2 N, i.e., all intervals Jk are nondegenerate
40 1
1
x
y
C1 C2
C3 C4
(a) Gluing four copulas
0 1
1
x
y
(b) Not realizable by consecutive
gluing
Figure 2: Compositions of horizontal and vertical gluing
2. There are degenerate intervals Jk = [ak;bk] with ak = bk.
Note that the second case can happen, as in the partition
I = [0;0] [
[
k¸1
h 1
k + 1
;
1
k
i
:
Let fCkg be a family of copulas Ck : I2 ! I with the same indexing
as fJkg. Then we de¯ne the function C : I2 ! I by
C(x;y) = (bk ¡ ak)Ck
³ x ¡ ak
bk ¡ ak
;y
´
+ aky
if x 2 [ak;bk] with bk ¡ak > 0, and extend it as a continuous function
to degenerate intervals with ak = bk. Then the same arguments as
above show that C is a bivariate copula. Note that the case of a ¯nite
partition can also be realized by sequentially applying the t
X
-operation.
Finally, we may combine compositions of horizontal and vertical
gluing. For instance, given four copulas C1;:::;C4, the copula
(C3 t
X
C4) t
Y
(C1 t
X
C2)
might be represented by the partition of I2 outlined in Figure 2. It is
worth mentioning, however, that not every partition of I2 into rect-
angles can be realized by consecutive gluing; an example of such a
con¯guration is shown in Figure 2.
50 1
1
x
y
q
Figure 3: The support of the singular copula C in Example 4.1
4 Examples
We illustrate the gluing construction with some examples. A copula
C is called singular if @2C
@x@y vanishes almost everywhere in I2; in this
case, the support of C has Lebesgue measure zero in I2. We refer the
reader to [3] for more details.
Example 4.1. Let µ 2 (0;1), and suppose that probability mass µ is
uniformly distributed along the line segment joining (0;0) and (µ;1),
and probability mass 1¡µ is uniformly distributed along the segment
between (µ;1) and (1;0). Consider the resulting singular copula C
whose support consists of these two line segments; see Figure 3. It
follows (see [3, Example 3.3]) that
C(x;y) =
8
> <
> :
x if x · µy
µy if µy < x < 1 ¡ (1 ¡ µ)y
x + y ¡ 1 if 1 ¡ (1 ¡ µ)y · x:
This copula is a standard example of a singular copula. In terms of
gluing, C can be written as
C = C+ t
X
C¡
where C+(x;y) = min(x;y) and C¡(x;y) = max(x + y ¡ 1;0) is the
Fr¶ echet-Hoe®ding upper and lower bound, respectively.
Example 4.2. Clearly, using the gluing construction one can combine
di®erent dependence relations on di®erent domains. In particular,
6gluing together an arbitrary copula C and the independence copula
P(x;y) = xy yields P t
X
C, respectively, C t
X
P. Note that P t
X
C = P
on [0;µ] £ I, and C t
X
P = P on [µ;1] £ I, respectively; in particular,
P t
X
P = P:
Acknowledgements
This work was partially supported by the German Science Foundation
(Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 475, Re-
duction of Complexity in Multivariate Data Structures). The second
author gratefully acknowledges a scholarship from the Ruhr Graduate
School in Economics and, in particular, from the Alfried Krupp von
Bohlen and Halbach Foundation.
References
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functions. Comm. Stat. Theory and Methods, 34(7):1437{1447,
2005.
[2] Gregory A. Fredricks, Roger B. Nelsen, and Jos¶ e Antonio
Rodr¶ ³guez-Lallena. Copulas with fractal supports. Insurance:
Mathematics & Economics, 37(1):42{48, 2005.
[3] R.B. Nelsen. An Introduction to Copulas. Springer, New York,
2nd edition, 2006.
[4] M. Sklar. Fonctions de r¶ epartition µ a n dimensions et leurs marges.
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7

An Introduction to Copulas.

and Jos¶ e Antonio Rodr¶ guez-Lallena. Copulas with fractal supports.

Fonctions de r¶ epartition µ a n dimensions et leurs marges.

New families of copulas based on periodic functions.

Gluing copulas

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