Copulas: theory and applications

New copulas based on general partitions-of-unity

and their applications to risk management

Prepared by Dietmar Pfeifer, Hervé Awoumlac Tsatdem,

Côme Girschig, Andreas Mändle

Presented to the ASTIN COLLOQUIUM

LISBOA 2016
31 MAY – 03 JUNE 2016

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 New copulas based on general partitions-of-unity
and their applications to risk management

Dietmar Pfeifer1, Hervé Awoumlac Tsatedem†,

Côme Girschig2, Andreas Mändle 1

Abstract: We construct new multivariate copulas on the basis of a

generalized infinite partition-of-unity approach. This approach allows - in
contrast to finite partition-of-unity copulas - for tail-dependence as well as for
asymmetry. A possibility of fitting such copulas to real data from quantitative
risk management is also pointed out.

Key words: copulas, partition-of-unity, tail dependence, asymmetry

1. Introduction

The theory of copulas and their applications has gained much interest in the recent
years, especially in the field of quantitative risk management, insurance and finance
(see e.g. MCNEIL, FREY AND EMBRECHTS (2005) or RANK (2006)). While classical
approaches like elliptically contoured copulas and Archimedean copulas are
widely explored, more modern approaches concentrate on non-standard, non-
symmetric or data-driven copula constructions (see e.g. LAUTERBACH AND PFEIFER
(2015), LAUTERBACH (2014), COTTIN AND PFEIFER (2014) or JAWORSKI, DURANTE AND HÄRDLE
(2013) and the papers therein for a survey). Statistical and computational aspects of
copulas have also been investigated in more detail recently (see e.g. BLUMENTRITT
(2012) and MAI AND SCHERER (2012)). In this paper, we want to focus on a particular
class of copulas and their generalizations, the so called partition-of-unity copulas
(see e.g. LI, MIKUSIŃSKI AND TAYLOR (1998) or KULPA (1999)). Whereas in the usual
approach, only finite partitions-of-unity are considered, which do not allow for a
modelling of tail-dependence, we extend this concept to infinite partitions-of-unity,
which allows for tail-dependence as well as for asymmetry, and which can also be
used to fit given data to a more realistic copula model. Our investigations resemble
in some sense more recent approaches such as YANG ET AL. (2015), GONZÁLEZ-BARRIOS
AND HERNÁNDEZ-CEDILLO (2013), ZHENG ET AL. (2011), HUMMEL AND MÄRKERT (2011), or
GHOSH AND HENDERSON (2009). Whereas in these papers, local modifications of known
standard copulas are considered in order to obtain tail dependence or
asymmetries, we focus on a closed form representation of completely new copula
densities which allows for easy Monte Carlo simulations as well as a data driven
modelling of tail dependence and asymmetries. This approach is not restricted to
two dimensions in general, but can likewise be used in arbitrary dimensions.
However, in order to illustrate our results, we will give examples in the bivariate case
only.

1 University of Oldenburg and 2 École Nationale des Ponts et Chaussées, Paris

2. Main Results

Let + = {0,1,2,3,} denote the set of non-negative integers and suppose that
{ji (u)}iÎ + and {y (v)}
j i Î+
are non-negative maps defined on the interval (0,1) each
such that

¥ ¥

å ji (u) = å yj (v) = 1
i =0 j =0
(2.1)

and
1 1

ò ji (u)du = ai > 0, ò y (v)dv = b

j j > 0 for i, j Î + . (2.2)
0 0

The maps ji (u) and yj (v) can be thought of as representing discrete distributions
over the non-negative integers + with parameters u and v, resp. The sequences
{ai }iÎ+ and {b j } + then represent the probabilities of the corresponding mixed
jÎ

distributions each.

Let further {p } ij i , j Î+

represent the probabilities of an arbitrary discrete bivariate
¥
distribution over + ´+ with marginal distributions given by pi  = å pij = ai and
j =0
¥
p j = å pij = b j for i, j Î + . Then
i =0

¥ ¥ pij
c(u, v):= å å ji (u)yj (v), u, v Î (0,1) (2.3)
i =0 j = 0 ai b j

defines the density of a bivariate copula, called generalized partition-of-unity

copula. The fact that c in fact is the density of a bivariate copula can be seen as
follows:

1 1
¥ ¥ pij ¥ ¥ pij
ò c(u, v)dv = å å ji (u) ò yj (v)dv = å å b j ji (u)
0 i =0 j = 0 ai b j 0 i =0 j =0 ai b j
¥ ¥ pij ¥
ji (u) ¥ ¥
ji (u) ¥
= åå ji (u) = å å p = å a = å ji (u) = 1, (2.4)
i =0 j = 0 ai i =0 ai j=0 ij i =0 ai i
i =0

likewise for ò c(u, v)du.

0

Note that from a „dual“ point of view, we can rewrite (2.3) as

¥ ¥
c(u, v) = åå pij fi (u) gj (v), u, v Î (0,1) (2.5)
i =0 j =0
 ji (  ) yj (  )
where fi (  ) = , gj (  ) = , i, j Î + denote the Lebesgue densities induced by
ai bj
{ji (u)}iÎ + and {y (v)}j iÎ+
. This means that the copula density c can also be seen as
an appropriate mixture of product densities, which possibly allows for a simple way
for a stochastic simulation.

An extension of this approach to d dimensions with d > 2 is obvious: assume that

{jki (u)}iÎ+ for k = 1, , d represent discrete probabilities with
¥

åj
i =0
ki (u) = 1 for u Î (0,1) (2.6)

and
1

òj ki (u)du = aki > 0 for i Î  + . (2.7)

0

Let further {pi }iÎ +d represent the distribution of an arbitrary discrete d-dimensional
random vector Z over +d where, for simplicity, we write i = (i1, , id ) , i.e.

P (Z = i) = pi , i Î +d . (2.8)

Suppose further that for the marginal distributions, there holds

P ( Zk = i ) = aki , i Î + , k = 1, , d. (2.9)

Then

pi d
d
c(u):= å d  jk ,ik (uk ), u = (u1, , ud ) Î (0,1) (2.10)
i Î + d
a
k=1
k , ik
k=1

defines the density of a d-variate copula, which is also called generalized partition-
of-unity copula.

Alternatively, we can rewrite (2.10) again as

d
d
c(u) = å pi  fk ,ik (uk ), u = (u1, , ud ) Î (0,1) (2.11)
iÎ+ d k=1

jki (  )
where the fki (  ) = , i Î + , k = 1,, d denote the Lebesgue densities induced by
aki
the {jki (u)}iÎ+ .
3. The symmetric case (diagonal dominance)

For simplicity, we restrict ourselves to the two-dimensional case in the sequel. The
generalization to higher dimensions is obvious.

Let ji = yi for i Î  and ò j (u)du = a > 0. Define

+
i i
0

ìïa , if i = j
pij := ïí i (3.1)
ïïî0, otherwise.

Then

¥
ji (u)ji (v) ¥
c(u, v):= å = å ai fi (u)fi (v), u, v Î (0,1) (3.2)
i =0 ai i =0

defines the density of a bivariate copula, called generalized partition-of-unity

copula with diagonal dominance.

Example 1 (binomial distributions - Bernstein copula). Consider, for a fixed integer

m ³ 2, the family of binomial distributions given by their point masses

ìïæm - 1ö i
ïç
ïç ÷÷ u (1- u)m-1-i , i = 0, , m - 1
jm,i (u) = íè i ø÷÷
ï ç (3.3)
ïï
ïïî0, i ³ m.

Here we have, for i = 0, , m - 1,

1
æm - 1ö÷ 1 i
am,i = ò jm,i (u)du = çç
çè i ø÷÷ ò
÷ u (1- u)m-1-i du
0 0

(m - 1)! G(i + 1)G(m - i) (m - 1)! i !(m - 1- i)! 1

= ⋅ = ⋅ = (3.4)
i !(m - 1- i)! G(m + 1) i !(m - 1- i)! m! m

and hence
2
æm - 1ö÷
m-1
cm(u, v) = må çç
m-1-i
÷÷ (uv)i ((1- u)(1- v)) , u, v Î (0,1) (3.5)
ç
i =0 è i ø ÷

which corresponds to the density of a particular Bernstein copula (see e.g. COTTIN
AND PFEIFER (2014), Theorem 2.1). Especially, for m = 2, we obtain

c2 (u, v) = 4uv - 2u - 2 v + 2, u, v Î (0,1). (3.6)

The corresponding copula C2 is given by

 x y

C2 (x, y) = ò ò c (u, v)dv du = xy + xy(1- x)(1- y),

2 x, y Î (0,1) (3.7)
0 0

and belongs to the so called Farlie-Gumbel-Morgenstern family (cf. e.g. NELSEN

(2006), p. 77). For general m >1, relation (3.5) represents the density of a copula with
polynomial sections of degree m in both variables (cf. NELSEN (2006), chapter 3.2.5).
The following graphs show some of these densities for different values of m.

m= 2 m= 3 m= 4 m= 5

Clearly, all those densities are bounded by the constant m, hence the coefficients
lU and lL of upper and lower tail dependence are zero:

1 1

òòc m (u, v)du dv

m(1- t)2
lU = lim t t
£ lim = 0,
t1 1- t t1 1- t
t t

òòc m (u, v)du dv

mt 2
lL = lim 0 0
£ lim = 0. (3.8)
t 0 t t 0 t

Example 2 (negative binomial distributions). Consider, for fixed b > 0, the family of
negative binomial distributions given by their point masses

æb + i - 1ö÷
jb ,i (u) = çç ÷÷(1- u)b ui , i Î  + . (3.9)
çè i ÷ø

Here we have, for i Î + ,

1
æb + i - 1÷ö 1 i G(b + i) G(i + 1)G(b + 1) b
ab ,i = ò ç
jb ,i (u)du = ç ÷÷ ò u (1- u)b du = ⋅ = (3.10)
çè i ÷
ø0 i ! G(b ) G(b + i + 2) (b + i)(b + i + 1)
0

and hence
 b
((1- u)(1- v)) ¥ æ b + i - 1÷ö
2

cb (u, v) = å (b + i)(b + i + 1)çç ÷ (uv)i

b i= 0
ç
è i ø÷÷
¥ æ
b + i - 1÷öæç b + i + 1ö÷
= (b + 1)((1- u)(1- v)) å çç
b
÷÷ ç ÷(uv)i , u, v Î (0,1). (3.11)
ç
i= 0 è i ÷ç
øè i ø÷÷

For integer choices of b, this expression can be explicitly evaluated as a finite sum,
as can be seen from the following result.

Lemma 1. For b Î , there holds

b
((1- u)(1- v)) b -1 æb - 1÷öæb + 1÷ö
cb (u, v) = (b + 1)
(1- uv)2 b +1
å çççè
i =0
÷÷çç
i øè i ÷ø
÷ ç
÷÷(uv)i , u, v Î (0,1). (3.12)

Proof. We will show by induction the equality of the following two expressions:

¥ æ
b + i - 1÷öæçb + i + 1ö÷ i 1 b -1 æ öæ ö
ççb - 1÷÷ççb + 1÷÷ z i for 0 < z < 1.
K(b , z):= å çç ÷÷ç ÷÷ z = k(b , z):= å
ç
i =0 è i ÷ç
øè i ÷ø (1- z)2 b +1
i=0 è
ç i ÷øèç i ø÷÷
÷
(3.13)

First notice that we have, for b Î  and 0 < z < 1,

¥ æ
¶K(b , z) b + i - 1öæ
÷÷ççb + i + 1ö÷÷ z i-1 and ¶ K(b , z) =
2 ¥ æb + i - 1öæ
÷÷ççb + i + 1ö÷÷ z i-2
= å i çç ÷÷øèç ÷÷ø å i(i - 1)çç
÷÷øèç ÷÷ø
¶z ç
i =1 è i i ¶z 2 i =2
çè i i

(3.14)

from which we can conclude the relation

¶ 2 K(b , z) ¶K(b , z)
z +
2
¶ K(b , z) 1 æç ¶K(b , z)ö÷ ¶z 2 ¶z .
= ççb(b + 2)K(b + 1, z) - ÷ or K(b + 1, z) = (3.15)
¶z 2 zè ¶z ÷ø b(b + 2)

A similar, but more elaborate calculation shows that the latter equality remains valid
if K(b , z) is replaced by k(b , z):

¶2 k(b , z) ¶k(b , z)
z +
k(b + 1, z) = ¶z 2 ¶z . (3.16)
b(b + 2)

In the first step of the induction, for b = 1, we have

¥
(i + 1)(i + 2) i ¥
j( j - 1) i-2 1 1
K(1, z) = å z =å z = h ''(z) = = k(1, z) (3.17)
i =0 2 j =2 2 2 (1- z)3
 ¥
1
with h(z):= å z i = for z <1. For the second step, assume that relation (3.13)
i=0 1- z
holds for some b Î . Then it follows by (3.15) and (3.16) that

¶ 2 K(b , z) ¶K(b , z) ¶ 2 k(b , z) ¶k(b , z)

z + z +
K(b + 1, z) = ¶z 2 ¶z = ¶z 2 ¶z = k(b + 1, z) (3.18)
b(b + 2) b(b + 2)

which finishes the proof. 

To give an illustration of Lemma 1, we show an exemplary table for some b, likewise

x y

for the corresponding copula Cb (x, y) = ò ò c (u, v)dv du,

b x, y Î (0,1).
0 0

b cb (u, v), u, v Î (0,1)

(1- u)(1- v)
1 2 3
(1- uv)
(1+ 3uv)(1- u)2 (1- v)2
2 3
(1- uv)5

(1+ 8uv + 6u v )(1- u) (1- v)

2 2 3 3

3 4
(1- uv)7

4
(1+ 15uv + 30u v 2 2
+ 10u3 v3 )(1- u)4(1- v)4
5
(1- uv)9

5
(1+ 24uv + 90u v 2 2
+ 80u3 v3 + 15u4 v 4 )(1- u)5(1- v)5
6
(1- uv)11

b
Cb (x, y), x, y Î (0,1)

(2 - x - y)
1 xy
1- xy

2
(3 - 3 x - 3 y + x 2
+ y 2 + 3 x2 y 2 - x2 y 3 - x3 y 2 )
xy
(1- xy)3

xy
(4 - 6x - 6y + 4 x2 + 4 xy + 4y 2 + 4 x3 y + 24 x2 y 2 + 4 xy 3 - x3 - 6 x2 y - 6 xy 2 - y 3 -
(1- xy)5
3
 - x 5 y 4 - x 4 y 5 + 4 x 4 y 4 - x 4 y 3 - x3 y 4 + 4 x4 y 2 + 4 x 3 y 3 + 4 x 2 y 4 - x4 y - 16 x3 y 2 - 16 x 2 y 3 -
The following graphs show the negative binomial copula densities cb for b = 1, ,4.

b =1 b=2 b=3 b=4

Negative binomial copulas typically show an upper tail dependence, as can be

seen from the following exemplary table.

b 1 2 3 4 5 6 7 8 9 10
1 5 11 93 193 793 1619 26333 53381 215955
lU(b )
2 8 16 128 256 1024 2048 32768 65536 262144

A closed formula for the tail dependence coefficients for integer values of b as a
finite sum is given in the following result.

Lemma 2. For b Î , there holds

1 1

ò ò c (u, v)du dv
b
2G(2b )
1 1
xb y b
G (b ) ò0 ò
lU(b ) = lim t t
= 2 ⋅ dx dy
t1 1- t 0
(x + y)2 b +1
2G(2b ) çæ b æçb ÷ö (-1)k b +k-2 æçb + k÷ö (-1)j +1 æç 1 ö÷÷ö
= 2 ⋅ ççå ç ÷÷ å èç j + 2 ÷ø÷ j + 1 èç 2 j+1 ø÷÷÷ø÷÷.
ç ÷ ç1- (3.19)
G (b ) çè çè k ÷ø b + k k=0 j =0

Proof. First, note that for b Î ,

b -1 æb - 1÷öæb + 1ö÷ b -1 æ b - 1 öæ
÷÷ççb + 1÷÷ö = ççæ 2b ÷÷ö
å ççèç
i =0
÷çç
÷ ÷÷ = å çç ÷
÷ç i ÷÷ø çèb - 1÷÷ø
i ÷øèç i ÷ø i=0 çèb - 1- iøè
(3.20)

which is a special case of Vandermonde’s identity. This in turn implies

b -1 æ
b - 1÷öæçb + 1÷ö æ 2b ÷ö 2G(2b )
(b + 1)å çç ÷÷ç ÷÷ = (b + 1)çç ÷= 2 . (3.21)
i=0 è
ç i øè ç
÷ i ø÷ èb - 1ø÷÷
ç G (b )
Now, in the light of Lemma 1, we obtain

1 1 1 1
(1- u)b (1- v)b
òò cb (u, v)du dv b -1 æ
b - 1÷öæçb + 1ö÷
òò (1- uv) 2 b +1
(uv)i du dv
lU(b ) = lim 1-h 1-h = å çç ÷÷ç ÷ lim 1-h 1-h
. (3.22)
h0 h ç i øè
i =0 è
÷ç i ø÷÷ h0 h

To evaluate the last integral, we substitute s = 1- u, w = 1- v and get

1 1 h h
(1- u)b (1- v)b sb wb
I(b , h, i):= ò ò (1- uv)2b +1 (uv) du dv = ò
i
ò
i
(1- s)(1- w)i ds dw. (3.23)
1-h 1-h 0 0
(s + w - sw)2 b +1

In a further step, substituting s = hx, w = hy , we obtain

1 1
xb y b
I(b , h, i) = hò ò
i
(1- hx)(1- hy)i dx dy , (3.24)
0 0
(x + y - hxy)2 b +1
giving

b -1 æ
b - 1öæ
÷÷ çç b + 1ö÷÷ lim I(b , h, i) =
b -1 æ öæ ö1 1

lU (b ) = å çç çç b - 1÷÷ çç b + 1÷÷ xb y b
i=0 è
ç i ÷÷øèç i ÷÷ø h0 h
å
i=0 è
ç i ÷øè ÷ ç i ÷ø÷ò ò (x + y )2 b +1
dx dy
0 0
1 1
2 G(2 b ) x y b b

G 2 (b ) ò0 ò (x + y )
= ⋅ 2 b +1
dx dy. (3.25)
0

It remains to evaluate the integral term in the expression above. Using the
substitution z = x + y and the binomial theorem twice, we get

1 1 1 y+1 1 y+1 b
xb y b (z - y)b æb÷ö k
çç ÷(-1)k y dz dy
ò ò (x + y)2b+1 dx dy = ò ò z2b+1 dz dy = ò y ò
y b b
å ç ÷÷
k=0 è kø zb+k+1
0 0 0 y 0 y

æbö÷ b +k 1 æ
b
ç k y çç 1 - 1 ö÷
= åç ÷÷(-1)
b + k ò0 èç yb +k (y +1)b +k ø÷÷
÷ dy
ç ÷
k=0 è kø

b (-1)k æç çæ y ÷ö ÷÷ö b +k 1 æ
1 ö÷ ö÷÷
b æ ö 1 b +k b æ ö b +k
b÷ k y çç æç
= åçç ÷÷÷ çç1-ç
ò çç èç y +1÷ø ÷÷÷ å ç
b + k ò0 çèç çè y +1÷ø ÷ø÷
÷ dy = ç ÷(-1) 1-
ç ç 1- ÷ ÷ dy
ç ÷
k=0 è kø b + k 0 è ø ç ÷÷
k=0 è k ø

b (-1)k çæ ö÷
b æ ö b +k æ 1
b + k÷ö j+1 1
= åçç ÷÷÷ çç(b + k)ln(2) + åçç ÷÷(-1) ò dy÷÷
ç ÷
k=0 è kø b + k ç
ç j ÷ø (y +1) j ÷
è j =2 è 0 ø÷
æ 1 ÷ö
ç 1-
b æ ö
b b (-1) ç çb + k÷ j+1 2 j-1 ÷÷÷
b æ ö k ç b +k æ ö
= ln(2)åçç ÷÷÷(-1)k + åçç ÷÷÷ ççåç ÷(-1) ÷
ç ÷
k=0 è k ø
ç ÷ ç ç j ø÷÷
k=0 è k ø b + k ç j=2 è j -1 ÷÷÷
ççè ÷÷
ø
b æ ö k b +k-2 æ ö
ççb + k÷÷ (-1) æçç1- 1 ö÷÷
j 1
b (-1) +
= åçç ÷÷÷ å ÷÷ (3.26)
ç ÷ ç
k=0 è k ø b + k j=0 è j + 2 ø j +1 è
ç 2 j+1 ø÷

b æb ö
since å çççè k ø÷÷÷÷(-1)
k=0
k
= (1- 1)b = 0, which proves the statement above. 
Example 3 (Poisson distributions). Consider the family of Poisson distributions given by
their point masses

g i L(u)i
jg ,i (u) = (1- u)g , i Î + (3.27)
i!

where L(u) = - ln(1- u) > 0, u Î (0,1) and g > 0. Here we get, for i Î + , with the
substitutions z = L(u) and y = (1+ g )z,

1 1 ¥
g i L(u)i g i z i -(1+g )z
ag , i = ò jg ,i (u)du = ò (1- u) g
du = ò e dz
0 0
i! 0
i!
¥ i
gi y i -y gi æ g ö÷ æ g ö÷
= i +1 ò
e dy = = çç ÷ çç1- ÷, (3.28)
(1+ g ) 0 i ! (1+ g )i +1
èç1+ g ø÷ èç 1+ g ø÷÷
÷

indicating that the ag ,i correspond to a geometric distribution with mean g, and

hence
i
¥
(g(1+ g)ln(1- u)ln(1- v))
cg (u, v) = (1+ g )(1- u) (1- v)
g g
å
i =0 i !2
, u, v Î (0,1). (3.29)

The following graphs show some of these copula densities for different choices of g.

g =1 g=2 g=5

g = 10 g = 20 g = 30
The corresponding copula C cannot be calculated explicitly. However, in contrast
to the visual impression, the coefficient lU(g ) of upper tail dependence is zero here
for all g > 0, although we have a singularity in the point (1,1) in all cases.

For a rigorous proof, we first remark that

¥
x i y i æç ¥ x i ÷ö æç ¥ y i ÷ö
h(x, y):= å £ ççå ÷÷ ⋅ ççå ÷÷ = exp(x + y) for all x, y ³ 0 (3.30)
i =0 i !2 çè i=0 i ! ÷ø èç i=0 i ! ÷ø

such that, with the constant K := g - g(1+ g ),

(
cg (u, v) = (1+ g )(1- u)g (1- v)g h - g(1+ g )ln(1- u), - g(1+ g )ln(1- v) )
£ 2(1- u)K (1- v)K , u, v Î (0,1). (3.31)

This implies
1 1

ò ò c (u, v)du dv g
(1- t)2 K +1
lU(g ) = lim t t
= 2lim = 0, b (3.32)
t 1 1- t t 1 (K + 1)2

as stated. (Note that 2K + 1= 1+ 2g - 2 g(1+ g ) > 0.)

Example 4 (log series distribution). Consider the family of log series distributions given
by their point masses

ui
ji (u) = , i Î + (3.33)
i ⋅ L(u)

where again L(u) = - ln(1- u), u Î (0,1). Here we get

1
1 i æç i ÷ö
ai = ò ji (u)du = å ç ÷÷÷(-1) ln( j + 1) for i Î .
ç
i j=1 è jø
j +1
(3.34)
0

The proof of this relation requires some more sophisticated arguments, as is shown in
the sequel.

Lemma 3. For c > 0 and n Î , there holds

n
¥
(1- e ) -x
n æ ö
n
ò e-cx dx = å çç ÷÷÷(-1)j +1 (ln( j + c) - ln(c)). (3.35)
x ç ÷
j =0 è j ø
0

n n
¥
(1- e ) -x
(1- e ) -x

Proof. Define gn(c):= ò e -cx

dx for c > 0. Note that fn(x):= for x > 0
0
x x
is bounded by 1 for all n Î . We can therefore apply the dominated convergence
theorem where appropriate. Now
 n
¥
(1- e ) -x ¥
n
¥ n ænö
gn '(c):= -ò
x
x ⋅ e-cx dx = -ò (1- e- x ) ⋅ e-cx dx = ò å çççè j ÷÷÷÷ø(-1)
j =0
j +1
e-( j +c)x dx
0 0 0

ænö ¥ n æ ö
n æ ¥ ö n æ ö
1 ççn÷÷(-1)j+1 1
n
ç
= å çç ÷÷÷(-1)j+1 ò e-( j +c)x dx = å çç ÷÷÷(-1)j +1 çç- ÷÷÷ =
ç ÷ ç ÷
e-( j+c)x ÷÷ å ÷÷ (3.36)
j =0 è j ø 0 j =0 è j ø
ç
çè j + c 0 ø j=0 çè j ø j+c

for c > 0. Let further

n æ ö
n
hn(c):= å çç ÷÷÷(-1)j +1 (ln( j + c) - ln(c)) for c > 0. (3.37)
ç ÷
j =0 è j ø

Then

n æ ö n æ ö æ 1
n d n 1ö
hn '(c) = å çç ÷÷÷(-1)j+1 (ln( j + c) - ln(c)) = å çç ÷÷÷(-1)j+1 çç - ÷÷÷
ç ÷
j =0 è j ø dc ç ÷
j =0 è j ø
çè j + c c ÷ø
n æ ö n æ ö
n æ 1 1ö n 1 1 n ænö
= å çç ÷÷÷(-1)j+1 çç - ÷÷÷ = å çç ÷÷÷(-1)j +1 + å çç ÷÷÷(-1)j
ç ÷
j =0 è j ø
çè j + c c ø÷ j=0 çè j ø÷ j + c c j=0 èç j ø÷
n æ ö
n 1
= å çç ÷÷÷(-1)j+1 (3.38)
ç ÷
j =0 è j ø j+c

n æ ö
n
since 0 = (1- 1)n = å çç ÷÷÷(-1)j . This implies gn ' = hn ' and hence gn(c) = hn(c) + Kn or
ç ÷
j=0 è j ø

equivalently, Kn = gn(c) - hn(c) for all c > 0, for some constant Kn Î . But then also

æ¥ n ö
çç (1- e-x ) -cx ÷÷ n æ ö
n
Kn = lim gn(c) - lim hn(c) = lim ç ò
ç e dx÷÷÷ - lim å çç ÷÷÷(-1)j+1 (ln( j + c) - ln(c))
c¥ c¥ c¥ ç x ÷ c¥ ç ÷
j =0 è j ø
èç 0 ø÷
n
¥
(1- e ) -x n æ ö
n æ æ j öö
¥ n
=ò lim (e-cx ) dx - å çç ÷÷÷(-1)j +1 lim ççlnçç1+ ÷÷÷÷÷÷ = ò 0 dx - å 0 = 0 (3.39)
x c¥ ç ÷
j =0 è j ø è èç cø÷ø 0
c¥ ç
j =0
0

for all n Î . Hence gn = hn for all n Î , which proves the Lemma. 

For the special case c = 1 we obtain, by the substitution x = - ln(1- u),

n
1
un
¥
(1- e-x ) -x n æ ö
n
bn := ò du = ò e dx = å çç ÷÷÷(-1)j+1 ln( j + 1). (3.40)
- ln(1- u) x ç ÷
j =1 è j ø
0 0

ui
Hence with ji (u) = - for i Î , this means
i ⋅ ln(1- u)
 1
bi 1 i æç i ÷ö
ai = ò ji (u)du = = å ç ÷÷(-1)j+1 ln( j + 1) for i Î . (3.41)
0
i i j=1 çè j÷ø

The density of the bivariate log series copula is hence given by

¥
1 1 ¥
(uv)i
c(u, v) = å ji (u)ji (v) = å for 0 < u, v < 1. (3.42)
i =1 ai ln(1- u)ln(1- v) i=1 ibi

The following graph shows the corresponding copula density.

plot of c(u, v)

The log series copula does not have a positive tail dependence either, as in the
case of the Poisson copula.
The proof of this statement again requires some more sophisticated arguments. We
proceed in the following steps.

Lemma 4. With L(u) = - ln(1- u), we have

1
1 L(ut)
lim
t 1 ò
1- t t L(u)
du = 1. (3.43)

Proof. Substitute s = 1- t. Then (3.43) is equivalent to

1 L (u ⋅ (1- s)) 1 L ((1- w)⋅ (1- s))

1 s

lim ò du = lim ò dw = 1, (3.44)

s0 s
1- s
L(u) s0 s
0
L(1- w)

with the substitution u = 1- w. This means that we have to show that

1 ln(w + s - ws)
s

s ò0
lim dw = 1. (3.45)
s0 ln(w)

Define
 ln(w + s) ln(w + s(1- s))
F(w, s):= , G(w, s):= . (3.46)
ln(w) ln(w)

Then

ln(w + s - ws)
F(w, s) £ £ G(w, s) for 0 < w £ s (3.47)
ln(w)

(note that ln(w) < 0 for 0 < w < 1). Now for 0 < s < 1,

æ s2 ÷ö
- ln ççç1- ÷÷
è w + s ÷ø
s s s
1 s ln(1- s)
0 £ ò G(w, s) - F(w, s)dw £ò dw £ - ln(1- s)ò dw £ (3.48)
0 0
- ln(w) 0
- ln(w) ln(s)

with the limit

s
1 ln(1- s)
0 £ lim
s0 ò
s 0
G(w, s) - F(w, s)dw £ lim
s0 ln(s)
= 0. (3.49)

Hence it suffices to prove

s s
1 1 - ln(w + s) !

s ò0 s0 s ò
lim F(w, s) dw = lim dw = 1. (3.50)
s0
0
- ln(w)

By the substitution x = -ln(w) we obtain the equivalent expression

1
¥
- ln (e- x + s) - x !
lim
s0 ò
s -ln( s) x
e dx =1. (3.51)

Note that

¥
- ln (e- x + s) ¥
- ln(e- x (1+ se- x )) ¥
x - ln (1+ se- x )
ò x
e- x dx = ò x
e- x dx = ò x
e- x dx
-ln( s) -ln( s) -ln( s)

¥ ¥
ln(1+ se- x ) ¥
ln (1+ se- x )
= ò e- x dx - ò x
e- x dx = s - ò x
e- x dx. (3.52)
-ln( s) -ln( s) -ln( s)

Hence it suffices to prove

1
¥
ln (1+ se- x ) - x !
lim ò e dx = 0. (3.53)
s0 s x
-ln( s)

With the substitution s = e-T , this is equivalent to

 ¥
ln(1+ ex-T ) !
lim eT ò e- x dx = 0. (3.54)
T ¥
T
x

Substituting finally y = x - T , this means

¥
ln(1+ ey ) ¥
ln(1+ ey ) !

ò dy = lim ò
T -(y + T )
lim e e e-y dy = 0. (3.55)
T ¥
0
y+T T ¥
0
y+T

But this is now evident due to

¥
ln (1+ ey ) ¥
y + 1 -y
¥
æ y + 1 -y ö÷
0 £ lim ò e-y dy £ lim ò e dy = ò lim çç e ÷÷ dy = 0 (3.56)
T ¥ y+T T ¥ y+T T ¥ ç
èy + T ø
0 0 0

by Lebesgue’s dominated convergence theorem. (For T ³1, an integrable majorant

is given by e-y .)

This proves Lemma 4. 

Lemma 5. With L(u) = - ln(1- u), the ai given in (3.34) and the copula density given in
(3.42), it holds that

1 1 1
1 t L(ut)
1- t òt ò 1- t òt L(u)
K(t):= c(u, v)du dv £ 1- du for 0 < t < 1, (3.57)
t

which in turn implies that the log series copula has no tail dependence.

æ 1 ö÷ 1 u
Proof. First notice that by the relation L(u) = - ln(1- u) = ln çç ÷£ - 1= for
çè1- u ø÷ 1- u 1- u
0 < u < 1, we obtain

1 1 1
ui 1 ui 1 1
ai = ò du ³ ò (1- u)du = ò ui-1(1- u)du = 2 for all i Î . (3.58)
0
i ⋅ L(u) i 0 u i 0 i (i + 1)

Now

1
1 1 ¥
(uv)i 1
1 ì
ï ¥
vi
1
ui ü
ï
ï ï dv
K(t):=
1- t òt ò å 2
i =1 ai i L(u)L(v)
du dv =
1- t òt
íå 2
ï a i L (v) ò L(u )
du ý
ï
(3.59)
t ï
î i =1 i t ï
þ

by Lebesgue’s dominated convergence theorem since for fixed v Î(0,1), by (3.58),

¥
(uv)i ¥
ui vi ¥
2 + (uv)2 - 3uv 3
å 2
i =1 ai i L(u)L(v)
£ å
i =1
(i + 1) ⋅ £ å
L(u) L(v) i=1
(i + 1)(uv)i -1
=
(1- uv) 3
£
(1- uv)3
, (3.60)
 1
3 3(2 - v)
the r.h.s being integrable w.r.t. u with value ò (1- uv) 3
du =
2(1- v)2
. Now for 0 < t < 1,
0

we have

t 1 1
ui vi ti vi
ò L(u)
du = t ò
L(vt )
dv ³ t i +1 ò
L(v )
dv (3.61)
0 0 0

and hence

1 1 t 1
ui ui ui vi
ò du = ò du - ò du £ (1- t i +1) ò dv = (1- t i +1) iai (3.62)
t
L(u) 0
L(u) 0
L(u) 0
L(v)

for all i Î . Thus we get, from (3.59),

1 ì
ï 1
vi
1
ui ü
ï 1 ¥
vi
ï dv £ 1
¥
ï
K(t) = íå 2
1- t òt
ï a i L (v) ò L(u )
du ý
ï 1- t ò å iL(v )
(1- ti+1) dv
ï
î i=1 i t ï
þ t i =1

1 ìïï ¥ vi (vt)i üïï

1 1 1
1 ¥
1 L(v) - tL(vt) t L(vt)
= ò íå - tå ý dv = ò dv = 1- ò dv, (3.63)
1- t t L(v) î ï
ï i=1 i i =1 i þ ï
ï 1- t t L(v) 1- t t L(v)

which proves relation (3.57). From Lemma 4 we thus obtain the final result

0 £ lU = lim K(t) £ 1- 1= 0 and hence lU = 0, (3.64)

t1

indicating that the log series copula has no upper tail dependence. 

4. The asymmetric case

Specifying the probabilities pij in a non-symmetric way we obtain asymmetric

copula densities even if the maps ji (  ) and yj (  ) are identical. A very simple
approach to this problem is a specification of a suitable non-symmetric
(n + 1)´(n + 1) -matrix Mn = éëê pij ùûú for n Î  + with
i , j=0,,n

n n

åp
k=0
ik = å pki = ai for i = 0, , n
k=0
(4.1)

and
ìïa , if i = j
pij := ïí i for i, j > n. (4.2)
ïïî0, otherwise
Example 5 (negative binomial distributions, asymmetric case). We consider the
negative binomial distributions from Example 2 with b = 1. Then
1
1
ai = ò j1,i (u)du = for i Î  + . With n = 4 and
0
(1+ i)(2 + i)

é18 5 5 0 2ù
ê ú
ê10 0 0 0 0úú
1 êê
M4 := 0 5 0 0 0úú (4.3)
60 êê
ê0 0 0 3 0úú
ê2 0 0 0 0úúû
êë

the conditions above are fulfilled, giving the copula density, according to (2.3),

¥ ¥ pij n n pij ¥
1
c(u, v) = å å ji (u)j j (v) = å å ji (u)j j (v) + å jk (u)jk (v), u, v Î (0,1) , (4.4)
i =0 j =0 ai a j i = 0 j =0 ai a j k= n+1 ak

or, more explicitly,

æ2 ö÷
çç ÷
çç6v ÷÷
çç ÷÷ ¥
(1- u)(1- v)
c(u, v) = (2 6u 12u2 20u3 30u4 ) ⋅ M4 ⋅ çç12v2 ÷÷÷ + å (k + 1)(k + 2)uk vk = H(u, v), u, v Î (0,1)
çç 3÷
÷ k= 5 5(1- uv)3
çç20 v ÷÷
ç ÷
çè30 v4 ø÷÷
(4.5)
with the polynomial

H(u, v) = 150u7 v7 - 450u6 v6 - 10u7 v3 + 510u5 v5 - 10u3 v7 - 30u5 v4 - 10u3 v5 + 30u2 v6 - 

 - 300u4 v4 + 30u6 v2 - 5u3 v4 + 80u4 v3 - 30u5 v + 94u3 v3 + 30u2 v4 - 30uv5 - 60u3 v2 + 
 + 15u2 v3 + 10u4 + 18u2 v2 - 30uv3 + 10 v4 - 15uv2 + 10 v2 - 18uv + 10u + 5v + 6. (4.6)

plot of c(u, v) plot of c(u, v) - c(v, u)

The corresponding copula C again has a coefficient of upper tail dependence
1
lU = as in the symmetric case.
2

The following example shows an asymmetric copula composed by two different

negative binomial distributions.

Example 6. We consider the negative binomial distributions from Example 2 with

1 1
1 2
b = 1 and b = 2. Then ai = ò j1,i (u)du = and b j = ò j2, j (v)dv =
0
(1+ i)(2 + i) 0
(2 + j)(3 + j)
= 2a j+1 for i, j Î  . Let further
+

ìïb j if j = 2i
ïï
ï
pij = íb j +1 if j = 2i + 1 for i, j Î + , (4.7)
ïï
ïï0 otherwise
î
i.e.

é b0 b1       ù
ê ú
ê  b2 b3     úú
ê
ép ù +
ëê ij ûú i , jÎ = êê    b4 b5   úú (4.8)
ê      b6 b7 úú
ê
ê        ú
êë úû

¥ ¥
where  stands for zero. Then pi  = å pij = ai and p j = å pij = b j for i, j Î + since
j =0 i =0

2 2 1
b2 i + b2 i +1 = + = = ai for i Î  + . (4.9)
(2 + 2i)(3 + 2 i) (3 + 2 i)(4 + 2 i) (1+ i)(2 + i)

It now follows from (2.5) that

¥ ¥
c(u, v) = åå pij fi (u) gj (v), u, v Î (0,1) (4.10)
i =0 j =0

(1- u)ui ( j + 1)(1- v)2 v j

is a copula density where fi (u) = and gj (v) = for
ai bj
i, j Î + , u, v Î (0,1). Using (4.8), one obtains, after some tedious but straightforward
calculations, that

2(1- u)(1- v)2 (1+ 2 v + 5uv 2 + 4uv 3 )

c(u, v) = 4
, u, v Î (0,1) (4.11)
(1- uv ) 2

which obviously is asymmetric.

 plot of c(u, v)

The corresponding copula C can again be calculated explicitly, giving

xy
C(x, y) = 4 (2 - x - 2 xy 3
+ xy 4 + x 2 y 3 - 2y 2 + y 3 ) , x, y Î (0,1). (4.12)
(1- xy )2

This copula has a coefficient of upper tail dependence

5
lU = (4.13)
9

which is between the coefficients of upper tail dependence for the symmetric case
with b = 1 and b = 2, cf. the final table in Example 2.

Remark 1: Negative binomial copulas (see Examples 2 and 5) can easily be

simulated through the alternative representation formula (2.5) involving mixed Beta
distributions here. Poisson copulas can be simulated using the transformation
z  1- e- z applied to Gamma distributed random variables Z with a random shape
parameter a, where a -1 is generated by the geometric distribution shown in (3.13),
and scale parameter 1+ g.

Remark 2: For practical applications in quantitative risk management, it seems

reasonable to fit the required probabilities êéë pij úùû + to empirical data via their
i , jÎ

empirical copula, for instance as was proposed in PFEIFER, STRASSBURGER AND PHILIPPS
(2009). In the particular case of Bernstein copulas (see Example 1) such a procedure
can be very easily implemented, even in higher dimensions (cf. COTTIN AND PFEIFER
(2014)).

As a practical exercise, we refer to Example 4.2 in COTTIN AND PFEIFER (2014) where the
empirical copula from an original data set was fitted to a Bernstein copula. The
following two graphs show the scatter plot from the empirical copula (big red dots)
superimposed by 1000 simulated points of that Bernstein copula (left) and of a
negative binomial copula of type (3.11), with b = 5.

Bernstein copula fit negative binomial copula fit

As can be nicely seen, the Bernstein copula represents the local asymmetry of the
empirical copula better, but shows no tail dependence, as does the negative
binomial copula.

The fit to the negative binomial copula was, for the sake of simplicity, performed by
a numerical match between the theoretical correlation for the negative binomial
copula and the correlation of the empirical copula, which is 0.815. Note that the
theoretical correlation r(b ) for the negative binomial copula of type (3.11) can be
explicitly calculated as

æ¥ (i + 1)2 ö÷
r(b ) = 12b çççå 2÷
÷ - 3 = 3b (2(b + 1)2 Y(1, b + 2) - 2b - 1) (4.14)
çè i=0 (b + i)(b + i + 1)(b + i + 2) ø÷

where Y(1, z) denotes the first derivative of the digamma function, or

d2
Y(1, z) = ln G(z), z > 0.
dz 2

b 1 2 3 4 5 6 7
r(b ) 0.4784 0.6529 0.7410 0.7937 0.8288 0.8537 0.8723

For the sake of completeness, we finally show a comparison between the Bernstein
copula fit and a Poisson copula fit with parameter g = 6. The empirical correlation
for the Poisson copula here is 0.814.
 Bernstein copula fit Poisson copula fit

Note that although the empirical plot for the Poisson copula might suggest some tail
dependence here this is actually not true in the light of (3.32).

More sophisticated fitting procedures - including asymmetric cases – also in higher

dimensions will be investigated elsewhere.

It should be finally pointed out that copula constructions as presented in this paper
will have a major impact in the construction of Internal Models under the new
Solvency II insurance supervising regime in Europe (see e.g. HUMMEL AND MÄRKERT
(2011) or SANDSTRǾM (2011), Chapter 13).

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