1. Multivariate Archimax copulas
Anne-Laure Fougeres
Institut Camille Jordan, Universite Lyon 1
joint work with
A. Charpentier, Ch. Genest and J.G. Neslehova
November 3, 2014
Workshop Extreme Value Theory, Spatial and Temporal Aspects
Besancon
2. I. Motivation
II. C ;L is a copula
III. Stochastic representations for Archimax copulas
IV. Simulation algorithms
V. Extremal behavior of Archimax copulas
VI. Conclusion - Perspectives
3. I. Motivation
Multivariate risks often deal with extremes of dependent variables:
I Alimentary risks: Global exposition to the contamination risk
via a set of aliments.
I Insurance risks: ruin probabilities, when several types of
contracts are concerned (natural disaster).
I Coastal
ooding: electrical infrastructures, dikes.
Multivariate extreme-value theory provides a useful
mathematical framework to handle such risks.
4. Consider a d-variate sample X1; : : : ;Xn, with Xi = (Xi1
; : : : ;Xid
),
for each i = 1; : : : ; n. De
5. ne
P(Xi x) = F(x) = C(F1(x1); : : : ; Fd (xd )) ;
so that F1; : : : ; Fd are the marginal cdfs (assume them continuous),
F is the joint cdf, and C is the associated copula.
Assumption: existence of a multivariate domain of attraction
There exist (an); (bn);G such that, when n ! 1,
Fn(an;1 x1 + bn;1; : : : ; an;d xd + bn;d ) = Fn(an x + bn) ! G(x);
where the attractor G is a d-variate cdf with non degenerate
margins G1; : : : ;Gd , and x is any continuity point of G.
6. This means equivalently that:
. the marginal cdfs Fj are in the univariate domain of attraction
of the Gj 's (j = 1; : : : ; d).
. there exists a d-variate copula C? such that for any u 2 [0; 1]d ,
lim
n!1
C(u1=n
1 ; : : : ; u1=n
d )n = C?(u1; : : : ; ud ) ; (1)
and the limiting cdfs are related via G(x) = C?(G1(x1); : : : ;Gd (xd )):
Notation: F 2 D(G) or C 2 D(C?).
Equation (1) is equivalent to
n
h
1 C
1
x1
n
; : : : ; 1
xd
n
i
! log C?(ex1 ; : : : ; exd ) = L?(x) :
L? = stable tail dependence function (Huang, 1992)
7. L?(x) = lim
n!1
n
h
1 C
1
x1
n
; : : : ; 1
xd
n
i
= lim
n!1
n P
h
F1(X1) 1
x1
n
or : : : or Fd (Xd ) 1
xd
n
i
:
Tail regions of interest for L?:
at least one of the components X1; : : : ;Xd becomes large.
8. Some properties of the stable tail dependence function L
I LM(x) := max(x1; : : : ; xd ) L(x) L(x) := x1 + + xd
comonotonicity case independence case
I margins are standardized: L(0; : : : ; 0; xj ; 0; : : : ; 0) = xj
I L is homogeneous of order 1
L(x) = lim
s!1
s
1 C
1
x1
s=
; : : : ; 1
xd
s=
= lim
t!1
t
h
1 C
1
x1
t
; : : : ; 1
xd
t
i
= L(x) :
I L is convex, i.e. for each 2 [0; 1],
Lfx + (1 )yg L(x) + (1 )L(y) :
10. x an attractor C? (equivalently L?); which kind of
distribution F does belong to its domain of attraction?
I theoretical descriptive interest.
I practical modeling interest: To get large families with a
exible structure in a speci
11. c domain of attraction.
I numerical interest: Risk evaluation requires estimation of C?.
Several estimators exist. How to compare them? For a small
sample simulation study, designs of experiments involve to .
12. x several attractors C?; . simulate, for each attractor, from several distributions
C 2 D(C?).
13. Attractor of classical multivariate distributions
I Multivariate normal d.f. ! Independence
I Archimedean copulas
C (u1; : : : ; ud ) =
1(u1) + + 1(ud )
where the generator : R+ ! [0; 1] satis
15. c conditions.
Archimedean copulas ! Multivariate logistic EV
L?(x1; : : : ; xd ) =
nPd
j=1 x1=r
j
or
(with r 1).
But in fact Independence case (r = 1) for
. Clayton's family (t) = (1 + t)1=; 0
. Frank's family (t) = log f1 et(1 e)g =
16. I Elliptical distributions
X = + RAU
where location parameter, R random radial component,
A d d-matrix invertible such that AAT positive de
17. nite, and
U random d-vector uniformly distributed on Sd1.
Elliptical distribution ! Independence
with R rapidly varying
I Extreme value d.f. CA ! itself CA
18. Objective: Construct a family of multivariate copulas
I which can have any extreme value distribution as its
maximum attractor
I which is easy to simulate.
Caperaa, Fougeres and Genest (2000) : bivariate Archimax copulas
C ;A(u1; u2) =
f 1(u1) + 1(u2)gA
1(u1)
1(u1) + 1(u2)
;
(2)
where A : [0; 1] ! [1=2; 1] and : [0;1) ! [0; 1] such that
(i) A is convex and, for all t 2 [0; 1], max(t; 1 t) A(t) 1;
(ii) : (0; 1] ! [0;1) is convex, decreasing, such that (0) = 1 and
limx!1 (x) = 0.
19. Archimax because... two important particular cases
I if A 1, C;A reduces to an Archimedean copula,
C (u1; u2) = f 1(u1) + 1(u2)g
I if (t) = et , C ;A is an extreme-value copula,
CA(u1; u2) = exp
ln(u1u2)A
ln(u1)
ln(u1u2)
:
Result: Archimax copulas are in the domain of attraction of an
EV copula CA? where, for all t 2 (0; 1),
A?(t) = ft1= + (1 t)1=gA
(
t1=
t1= + (1 t)1=
)
whenever t7! 1(1 1=t) is regularly varying of degree
1= with 2 (0; 1];
20. Additional references
I Application in hydrology: see Basigal, Jagr and Mesiar (2011);
I Applications in
21. nance: see Zivot and Wang (2006), Jaworski,
Durante and Hardle (2013), Mai and Scherer (2014);
I R package: acopula (Basigal);
I Mesiar and Jagr (2013): conjecture that a suitable extension
to arbitrary dimension should be
C ;L(u1; : : : ; ud ) = Lf 1(u1); : : : ; 1(ud )g: (3)
Open problem 4.1 (Mesiar and Jagr, 2013) : C ;L is a copula as
soon as L is a stable tail dependence function and is an
Archimedean generator.
[sounds reasonable, since for d = 2, A(t) = L(t; 1t), so that (3) is (2).]
22. Purpose of our work
I prove that C ;L is a copula
solve Open problem of Mesiar and Jagr, 2013
I study the d-variate Archimax family
. in terms of attractor;
. in terms of simulation issues.
Refer to Charpentier, Fougeres, Genest and Neslehova (2014),
JMVA 126, pp. 118-136.
23. C ;L is a copula: main ingredients
For all u1; : : : ; ud 2 (0; 1), consider
C (u1; : : : ; ud ) = f 1(u1) + + 1(ud )g:
1. Characterization of an Archimedean generator
Then C is a copula if and only if : [0;1) ! [0; 1] satis
24. es
. (0) = 1,
. limx!1 (x) = 0
. is d-monotone, i.e. has d 2 derivatives on (0;1),
(1)j (j) 0 (for all j 2 f0; : : : ; d 2g), and (1)d2 (d2)
non-increasing and convex on (0;1).
McNeil and Neslehova (2009)
25. C ;L is a copula: main ingredients (cont.)
2. Characterization of a stable tail dependence function [stdf]
L : [0;1)d ! [0;1) is a d-variate stdf if and only if
(a) L is homogeneous of degree 1;
(b) L(e1) = = L(ed ) = 1;
(c) L is fully d-max decreasing, i.e., for any J f1; : : : ; dg of
arbitrary size jJj = k and all x1; : : : ; xd ; h1; : : : ; hd 2 [0;1),
X
(1)1++k L(x1+1h1112J ; : : : ; xd+dhd1d2J ) 0:
1;:::;k2f0;1g
Ressel (2013)
27. ned by
f (y1; : : : ; yd ) = L(y1; : : : ;yd ) (4)
is totally increasing as de
28. ned in Morillas (2005), which states
X
(1)k1k f (y1+1h1112J ; : : : ; yd +dhd1d2J ) 0:
1;:::;k2f0;1g
29. First result
Theorem
Let L be a d-variate stdf and be the generator of a d-variate
Archimedean copula. There exists a vector (X1; : : : ;Xd ) of strictly
positive random variables such that, for all x1; : : : ; xd 2 [0;1),
Pr(X1 x1; : : : ;Xd xd ) = L(x1; : : : ; xd ):
In particular, Pr(Xj xj ) = (xj ) for xj 2 [0;1) and j 2 f1; : : : ; dg.
30. Sketch of proof:
I Morillas (2005) - McNeil and Neslehova (2009) :
Archimedean generator , y absolutely monotone of order d
where y : t 2 (1; 0]7! (t) 2 [0; 1].
I Morillas (2005) + (c) ) y f totally increasing.
I L satis
31. es (b)
) y f (y1; 0; : : : ; 0) = y[L(y1; 0; : : : ; 0)] = y(y1):
I continuous ) y f continuous.
Consequence: y f is a cdf on (1; 0]d . This means
equivalently that y f (x1; : : : ;xd ) = L(x1; : : : ; xd ) is a
survival function on [0;1)d .
32. Corollary
Let L be a d-variate stable tail dependence function and be the
generator of a d-variate Archimedean copula. Then
C ;L(u1; : : : ; ud ) = Lf 1(u1); : : : ; 1(ud )g
is a copula, as conjectured by Mesiar and Jagr (2013).
33. Some examples
I Recall that L(x) := x1 + + xd . Then C ;L is the
d-variate Archimedean copula C .
I If (t) = et , C ;L is the extreme-value copula with stdf L
C ;L(u1; : : : ; ud ) = exp[Lf ln(u1); : : : ;ln(ud )g] :
I Let 1 and consider the stdf of the d-variate logistic
extreme-value copula
L(x1; : : : ; ud ) = (x
d )1=:
1 + + x
Then for any generator ,
C ;(u1; : : : ; ud ) =
h
f 1(u1)g + + f 1(ud )g
i1=
is an Archimedean copula with generator (t) = (t1=).
34. III. Stochastic representations for Archimax copulas
1. is a Laplace transform. Suppose that is the Laplace
transform of a strictly positive r.v. with cdf G, so that
(t) =
Z 1
0
et dG():
Bernstein's Theorem (Widder, 1941) ) is completely monotone,
i.e., it is dierentiable of any order and for all k 2 N, (1)k (k) 0.
Let L be a d-variate stdf. Let (T1; : : : ;Td ) be a random vector
with survival function
Pr(T1 t1; : : : ;Td td ) = expfL(t1; : : : ; td )g: (5)
This means T1; : : : ;Td E(1) with survival copula the
extreme-value copula with stable tail dependence function L.
35. Stochastic representations for Archimax copulas (cont.)
Theorem
The copula C ;L is Archimax with d-variate stdf L and completely
monotone Archimedean generator if and only if it is the survival
copula of the random vector
(X1; : : : ;Xd ) = (T1=; : : : ;Td=) ;
where has Laplace transform and is independent of the
random vector (T1; : : : ;Td ) de
36. ned in (5).
Sketch of proof:
Pr(X1 x1; : : : ;Xd xd ) =
Z 1
0
Pr(T1 x1; : : : ;Td xd ) dG()
=
Z 1
0
expfL(x1; : : : ; xd )g dG()
= L(x1; : : : ; xd ):
37. Stochastic representations for Archimax copulas (cont.)
2. d-monotone. Consider 0(t) = max(0; 1 t)d1 (t 0).
0 is d-monotone ) there exists (S1; : : : ; Sd ) such that,
Pr(S1 s1; : : : ; Sd sd ) = [maxf0; 1 L(s1; : : : ; sd )g]d1:
Then
I support of this joint survival function:
d (`) = f(s1; : : : ; sd ) 2 [0; 1]d : L(s1; : : : ; sd ) 1g
I S1; : : : ; Sd are (dependent) Beta r.v. B(1; d 1).
Now let R be a strictly positive r.v. with cdf F, independent of
(S1; : : : ; Sd ) and consider
(X1; : : : ;Xd ) = (RS1; : : : ; RSd ): (6)
38. Stochastic representations for Archimax copulas (cont.)
Theorem
(i) If (X1; : : : ;Xd ) has form (6), then its survival copula is the
Archimax copula C ;L, where is the Williamson d-transform
of R, i.e., for all t 2 [0;1),
(t) =
Z 1
t
1
t
r
d1
dF(r ):
(ii) Let L be a d-variate stdf and be a generator of a d-variate
Archimedean copula. Then C ;L is the survival copula of a
random vector (X1; : : : ;Xd ) of the form (6), where the cdf F
of R is the inverse Williamson d-transform of ,
F(r ) = 1
dX2
k=0
(1)k r k (k)(r )
k!
(1)d1r d1 (d1)
+ (r )
(d 1)!
;
where (d1)
+ denotes the right-hand derivative of (d2).
39. Corollary
A function L : [0;1)d ! [0;1) is a d-variate stdf if and only if
(a) L is homogeneous of degree 1;
(b) The function given, for all x1; : : : ; xd 2 [0;1), by
G
`(x1; : : : ; xd ) = [maxf0; 1 L(x1; : : : ; xd )g]d1 (7)
de
40. nes a d-variate survival function with B(1; d 1) margins.
Remark: The distribution in (7) is related to the multivariate
generalized Pareto distribution of Falk and Reiss (2005). See also
Hofmann (2009).
41. Reminder: Purpose of our work
I prove that C ;L is a copula
solve Open problem of Mesiar and Jagr, 2013
I study the d-variate Archimax family
. in terms of simulation issues;
. in terms of attractor.
42. IV. Simulation algorithms
Algorithm 1
Let L be the d-variate stdf associated to an extreme-value copula
D, and let be a d-variate Archimedean copula generator.
Suppose that is the Laplace transform of a r.v. .
To simulate an observation (U1; : : : ;Ud ) from a d-variate
Archimax copula C ;L, proceed as follows:
1.1 Generate an observation (V1; : : : ;Vd ) from copula D.
1.2 Set T1 = ln(V1); : : : ;Td = ln(Vd ).
1.3 Generate an observation .
1.4 Set U1 = (T1=); : : : ;Ud = (Td=).
43. Algorithm 2
Let L be a d-variate stdf, and let be a d-variate Archimedean
copula generator.
To simulate an observation (U1; : : : ;Ud ) from C ;L:
2.1 Generate an observation (S1; : : : ; Sd ) from the joint survival
function de
44. ned, for all s1; : : : ; sd 2 [0;1), by
G
`(s1; : : : ; sd ) = [maxf0; 1 L(s1; : : : ; sd )g]d1:
2.2 Generate R from the cdf de
45. ned, for all r 2 (0;1), by
F(r ) = 1
dX2
k=0
(1)k r k (k)(r )
k!
(1)d1r d1 (d1)
+ (r )
(d 1)!
:
2.3 Set U1 = (RS1); : : : ;Ud = (RSd ).
46. V. Extremal behavior of Archimax copulas
Let X1;X2; : : : be iid copies of a vector X = (X1; : : : ;Xd ) whose
distribution is the Archimax copula C ;L, and de
47. ne for each n 2 N,
Mn = max(X1; : : : ;Xn);
where vector algebra is meant component-wise.
Objective:
48. nd the limiting behavior, as n ! 1, of the sequence (Mn).
Reminder for equation (1):
lim
n!1
C ;L(u1=n
1 ; : : : ; u1=n
d )n = C?(u1; : : : ; ud ) :
49. Extremal behavior of Archimax copulas (cont.)
Theorem
Suppose that is the generator of a d-variate Archimedean copula
such that w7! 1 (1=w) is regularly varying of index for
some 2 (0; 1]. Then the copula C ;L belongs to the maximum
domain of attraction of an extreme-value distribution whose unique
underlying copula is de
50. ned, for all u1; : : : ; ud 2 (0; 1), by
CL?(u1; : : : ; ud ) = exp[Lfj ln(u1)j1=; : : : ; j ln(ud )j1=g]:
51. VI. Conclusion - Perspectives
Our purpose has been to
I prove that C ;L is a copula
as conjectured by Mesiar and Jagr, 2013
I study the d-variate Archimax family
. in terms of simulation issues;
. in terms of attractor.
Some questions remains to be considered:
I computational issues associated with Algorithms 1 and 2.
I dependence structure.
For d = 2, Caperaa, Fougeres and Genest (2000) :
;L = L + (1 L) :
How to extend this relation to the multivariate case ?
52. References (1/2)
. T. Bacigal, V. Jagr, R. Mesiar (2011), Non-exchangeable random variables,
Archimax copulas and their
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extreme value attractor, J. Multivariate Anal. 72, 30-49. . A. Charpentier, A.-L. Fougeres, C. Genest, J.G. Neslehova (2014), Multivariate
Archimax copulas, J. Multivariate Anal. 126, 118-136. . M. Falk, R.-D. Reiss (2005), On Pickands coordinates in arbitrary dimensions, J.
Multivariate Anal. 92 426-453. . A.J. McNeil, J. Neslehova (2009), Multivariate Archimedean copulas,
d-monotone functions and `1-norm symmetric distributions, Ann. Statist. 37,
3059-3097. . J. F. Mai, M. Scherer (2014) Financial Engineering with Copulas Explained,
Palgrave Macmillan. . D. Hofmann (2009), Characterization of the D-norm corresponding to a
multivariate extreme value distribution, Ph.D.Thesis, Bayerische
Julius-Maximilians-Universitat Wurzburg, Germany.
54. References (2/2)
. X. Huang (1992), Statistics of bivariate extreme values, Ph.D. Thesis,
Tinbergen Institute Research Series, The Netherlands. . P. Jaworski, F. Durante, W. K. Hardle (2013) Copulae in Mathematical and
Quantitative Finance, Lecture Notes in Statistics, Vol. 213. Springer. . R. Mesiar, V. Jagr (2013), d-dimensional dependence functions and Archimax
copulas, Fuzzy Sets and Systems 228, 78-87. . P.M. Morillas (2005), A characterization of absolutely monotonic functions of a
55. xed order, Publ. Inst. Math. (Beograd) (N.S.) 78 (92), 93-105. . P. Ressel (2013), Homogeneous distributions - and a spectral representation of
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