Table Of ContentOn estimating extremal dependence structures
by parametric spectral measures
3
1
0 Jan Beran1 and Georg Mainik2
2
n
a 1Department of Mathematics and Statistics,
J
University of Konstanz
7
2 2RiskLab, Department of Mathematics, ETH Zurich
]
E
January 29, 2013
M
.
t
a
t Abstract
s
[
Estimation of extreme value copulas is often required in situations
1
where available data are sparse. Parametric methods may then be the
v
6 preferred approach. A possible way of defining parametric families
7
that are simple and, at the same time, cover a large variety of mul-
3
tivariate extremal dependence structures is to build models based on
6
. spectral measures. This approach is considered here. Parametric fam-
1
0 ilies of spectral measures are defined as convex hulls of suitable basis
3 elements, and parameters are estimated by projecting an initial non-
1
parametric estimator on these finite-dimensional spaces. Asymptotic
:
v
distributions are derived for the estimated parameters and the result-
i
X ing estimates of the spectral measure and the extreme value copula.
r Finite sample properties are illustrated by a simulation study.
a
1 Introduction
Extreme value copulas provide a suitable general approach to modelling mul-
tivariate extremes. Various nonparametric methods for estimating extreme
value copulas have been proposed in the last few years [17], [10], [11], [3]
(also see [15], [5] and [13] for related approaches). In practical applications,
1
such as for instance operational risk or rare natural disasters, one is however
often in a situation where available data are sparse. Nonparametric methods
generally require a fairly large sample size in order to be reliable. For small
samples and in situations where one may have some idea about plausible
properties of the distribution, parametric methods are likely to yield more
accurate results. An approach to parametric inference for extreme value
copulas is discussed for instance in [2].
One of the key issues is how to define parametric families that are simple
and at the same time general enough to cover a large variety of multivariate
dependence structures in the extremes. For instance, some of the most pop-
ular models are based on Archimedean copulas, which all correspond to the
same type of extremal dependence structure, characterized by the Gumbel
copula [7]. One way of achieving more flexibility in the extremes is to build
models based on spectral measures. This is the approach taken here. For
related work see e.g. [6], [12], and [11].
More specifically, the idea pursued in the following is to select a finite
number of suitable spectral measures as basis elements and to use their con-
vex combinations as a parametric family of dependence structures. Given a
sufficiently large number of such basis elements, any spectral measure can be
approximated by a weighted sum. Estimation of the coefficients can then be
carriedoutbyprojectinganonparametricestimator, suchastheonein[3], on
the finite-dimensional space generated by the basis elements. If the number
of basis elements in the model is large (and increasing with the sample size),
then projecting the original non-parametric estimator can be considered as
a discretization technique. This is the setting in [6] and [11].
On the other hand, an appropriate model with a small number of basis
elements can have the advantage of dimension reduction. Given a reason-
ableparametricmodelwithasmallnumberofparameters, onecanreducethe
variabilityofanonparametricestimatorbyprojectingitonalow-dimensional
space. This is the approach studied here. We define explicit parameter esti-
mators in the low-dimensional setting and study the asymptotic distribution
of the resulting estimators of the dependence structure. To illustrate the
potential advantage of dimension reduction, we construct an example with
three basis elements and compare a non-parametric estimator with its low-
dimensional projection in a simulation study.
Note that in principle any nonparametric estimator (cf. [5], [3], [13], [17],
[10], [11]) can be used as a starting point. Depending on the nonparametric
method used in the projection, the marginal distributions are either known
2
or estimated from the observed data. The asymptotic results given below
only require that a functional limit theorem in a suitable topology holds for
the initial estimator.
Thepaperisorganizedasfollows. Basicdefinitionsandconceptsofmulti-
variateextremevaluetheoryaresummarizedinsection2. Parametricmodels
in the spectral domain and a corresponding parametric estimator are intro-
duced in section 3. Asymptotic results, including consistency and a central
limit theorem, are derived in section 4. The theoretical results are illustrated
by simulations for a specific model in section 5. Final remarks in section 6
with a discussion of some open problems conclude the paper.
2 Basic definitions
ConsiderasampleX ,...,X consistingofiidrealizationsX = (X ,...,X )T
1 n i i,1 i,d
of a d-dimensional random vector X = (X ,...,X )T ∈ Rd with marginal
1 d
distributions F ,...,F and copula C . That is, F (t) = P (X ≤ t) for
1 d X j j
j = 1,...,d and t ∈ R, and
P (X ≤ x) = C (F (x ),...,F (x ))
X 1 1 d d
for x ∈ Rd. The notation x ≤ y for x,y ∈ Rd means x ≤ y for j = 1,...,d.
j j
The transposition operator (·)T in X = (X ,...,X )T indicates that X is
1 d
considered as a column vector. Distinguishing columns and rows will be
useful in some calculations later on.
The vector M = (M ,...,M )T of componentwise maxima
n n,1 n,d
M = max X
n,j i,j
i=1,2,...,n
then has marginal distributions P(M ≤ t) = Fn(t) and a copula C (u)
n,j j Mn
given by
P (M ≤ x) = C (Fn(x ),...,Fn(x )) = Cn (F (x ),...,F (x )).
n Mn 1 1 d d X 1 1 d d
In the limit one obtains, under general conditions, an extreme value copula
C(u) = lim Cn (cid:0)u1/n(cid:1) (u = (u ,...,u )T ∈ [0,1]d) with the characteris-
n→∞ X 1 d
tic max-stable property
(cid:0) (cid:1)
C(u) = Cn u1/n (1)
3
for all n ∈ N. For an accessible introduction to this topic see e.g. [9] and
references therein. The definition (1) of extreme value or max-stable copulas
is equivalent to the representation
C(u) = exp(−(cid:96)(−logu ,...,−logu ))
1 d
with the tail dependence function
(cid:90)
(cid:96)(x) = max (w x )dΨ(w ,...,w ), (x ∈ [0,∞)d), (2)
i i 1 d
i=1,...,d
∆
d
and Ψ the so-called spectral measure on the unit simplex in Rd, ∆ = {x ∈
d
[0,1]d : (cid:80)d x = 1}, satisfying
i=1 i
(cid:90)
w dΨ(w ,...,w ) = 1, (i = 1,...,d) (3)
i 1 d
∆
d
(cid:82)
(cf. Theorem 6.2.2 in [9]). Note that the last condition implies dΨ(w) =
∆
d
d. The underlying original results go back to [4] and [15]. The main conclu-
sion is that spectral measures, tail dependence functions, and extreme value
copulas are equivalent representations of dependence structures in multivari-
ate extreme value theory. Note also that (cid:96)(rx) = r(cid:96)(x) for r > 0 and
x ∈ [0,∞)d, so that it is sufficient to specify (cid:96)(x) for x ∈ ∆ only. The re-
d
striction of (cid:96) to ∆ is also called Pickands dependence function and is usually
d
denoted by A(·). The extension to x ∈ [0,∞)d is obtained by
(cid:18) (cid:19)
x
(cid:96)(x) = (cid:107)x(cid:107) (cid:96) = (cid:107)x(cid:107) A(w) (4)
1 1
(cid:107)x(cid:107)
1
where (cid:107)x(cid:107) = x + ··· + x (since x ≥ 0), and w = x/(cid:107)x(cid:107) . Note that
1 1 d j 1
condition (3) is equivalent to
A(e ) = 1, (j = 1,...,d) (5)
j
where e = (e ,...,e )T is the j-th unit vector in Rd: e = 0 (j (cid:54)= l) and
j j,1 j,d j,l
e = 1. That is, (3) standardizes A on the vertices of the unit simplex ∆ .
j,j d
4
3 Parametric models for spectral measures:
construction and estimation
3.1 Models
One way of building parametric models that encompass a large variety of
extremaldependencestructuresistostartatthelevelofthespectralmeasure
Ψ. Thus, suppose that Ψ ...,Ψ are some fixed spectral measures. The
1 p
correspodingdependencefunctionsandextremevaluecopulaswillbedenoted
by (cid:96) ,...,(cid:96) , A ,...,A and C ,...,C , respectively. A parametric family of
1 p 1 p 1 p
spectral measures P = {Ψ(·,θ),θ ∈ Θ}, and corresponding families A and
p p
C of (Pickands) dependence functions and copulas respectively, can then be
p
obtained by defining spectral measures of the form
(cid:32) (cid:33)
p−1 p−1
(cid:88) (cid:88)
Ψ(·,θ) = θ Ψ (·)+ 1− θ Ψ (·)
i i i p
i=1 i=1
where θ = (θ ,...,θ ) ∈ Θ and Θ = {ϑ ∈ (0,1)p−1 : (cid:80)p−1ϑ ≤ 1}. As
1 p−1 i=1 i
(3) remains valid for convex combinations, Ψ(·,θ) is a spectral measure by
definition. In terms of the corresponding dependence functions we have
(cid:32) (cid:33)
p−1 p−1
(cid:88) (cid:88)
(cid:96)(x,θ) = θ (cid:96) (x)+ 1− θ (cid:96) (x),
i i i p
i=1 i=1
(cid:32) (cid:33)
p−1 p−1
(cid:88) (cid:88)
A(w,θ) = θ A (w)+ 1− θ A (w).
i i i p
i=1 i=1
For the copulas we obtain
(cid:40) (cid:41)
p−1
(cid:88)
C(u,θ) = exp − θ [(cid:96) (−logu)−(cid:96) (−logu)]−(cid:96) (−logu) (6)
i i p p
i=1
(cid:32) (cid:33)
(cid:89)p−1(cid:18)C (u)(cid:19)θi
i
= C (u)
p
C (u)
p
i=1
where (cid:96) (−logu) = (cid:96) (−logu ,...,−logu ).
i i 1 d
Henceforth we assume that the parameter θ is identifiable in the sense
that A = A implies θ = θ(cid:48) for θ,θ(cid:48) ∈ Θ. A sufficient criterion for the
θ θ(cid:48)
5
identifiability of θ is linear independence of the basis elements A ,...,A . If
1 p
ˆ
θ is not identifiable, then an estimator θ may fail to converge. An important
example of this issue is the decomposition of discrete spectral measures.
According to [14], any discrete spectral measure on ∆ can be expressed as a
2
convex combination of two-point spectral measures. This result can also be
written in terms of piecewise linear dependence functions and Marshall-Olkin
copulas. However, the decomposition is not necessarily unique. This can be
illustrated by the following example. Let (t,1 − t) ∈ ∆ be represented by
2
the first coordinate t ∈ [0,1] and consider the family Q of discrete spectral
4
measures Ψ = (cid:80)4 c δ with c ≥ 0 for i = 1,...,4. It is easy to see
i=1 i (i−1)/3 i
that a basis of 2-point spectral measures needed for the decomposition of all
Ψ ∈ Q must include all elements of Q with only two atoms. These are
4 4
1 3 3 1
Ψ = δ +δ , Ψ = δ +δ , Ψ = δ + δ , Ψ = δ + δ .
1 0 1 2 1/3 2/3 3 0 2/3 4 1/3 1
2 2 2 2
The non-uniqueness follows from 1Ψ + 3Ψ = 1(Ψ +Ψ ).
4 1 4 2 2 3 4
3.2 Estimation
Several nonparametric estimators of Pickands dependence functions, spec-
tral measures, and corresponding extreme value copulas have been proposed
in the recent literature [5], [3], [13], [17], [10], [11]. Generally, these meth-
ods require fairly large sample sizes in order to achieve a sufficient degree of
accuracy. In contrast, parametric estimates are expected to be reasonably
accurate for moderate or even small sample sizes, provided that the paramet-
ric assumptions are sufficiently realistic. To see how much may be gained by
parametric estimation, we consider the following approach. Suppose that a
nonparametric estimate A(cid:98) of the dependence function A is given, and recall
that the corresponding (cid:98)l is obtained from A(cid:98) according to (4). A natural
parametric estimator based on the family A , and A(cid:98) as intial estimate, is
p
obtained by projecting the function A(cid:98) on A . Note that even if (2) does not
p
holdforA(cid:98)(seee.g. [10]), itholdsautomaticallyfortheprojectionofA(cid:98)onA ,
p
so that this projection is a proper dependence function by definition. Note
also that the projection improves the accuracy of the estimate if the true
dependence function is indeed in A . Related improvements for projections
p
on infinite-dimensional spaces of spectral measures and approximations by
sieve methods have been considered in [6].
6
Specifically, we may start for instance with the following nonparametric
estimator A(cid:98)considered in [10], [17], [3], [5] and [15]. Suppose that the depen-
dence structure of X ∈ Rd (i = 1,...,n) is characterized by an extreme value
copula C(·,θ) ∈ C , and, as before, the marginals are denoted by F ,...,F .
p 1 d
Given n iid realizations X (i = 1,2,...,n), define Y = (Y ,...,Y )T by
i i i,1 i,d
Y = −logF (X ).
i,j j i,j
Then Y are standard exponential random variables, and
i,j
(cid:0) (cid:1)
P (Y > y ,...,Y > y ) = C e−y1,...,e−yd,θ = exp(−(cid:96)(y,θ))
i,1 1 i,d d
= exp(−(cid:107)y(cid:107) A(w,θ))
1
for y ∈ [0,∞)d (and w = y/(cid:107)y(cid:107) ∈ ∆ ). Hence, for
1 d
Y
i,j
ξ (w) := min (i = 1,...,n)
i
j=1,..,d wj
with w ∈ ∆ one obtains
d
P(ξ (w) > t) = P (Y > w t,...,Y > w t) = exp(−tA(w)).
i i,1 1 i,d d
This means that, for any fixed w ∈ ∆ , ξ (w),...,ξ (w) are iid exponen-
d 1 n
tially distributed with mean 1/A(w), and −log(ξ (w)) (i = 1,...,n) are iid
i
Gumbel distributed with location parameter logA(w). In particular,
E(−logξ (w)) = logA(w)+γ
i
where γ is the expectation of the standard Gumbel distribution (i.e. γ =
Γ(cid:48)(1) ≈ 0.5772, the Euler-Mascheroni constant). A nonparametric estimator
of A may therefore be defined by (see [15], [5], [13], [6], [10])
n
1 (cid:88)
logA(cid:98)(w) = − logξ (w)−γ (w ∈ ∆ ). (7)
i d
n
i=1
Unfortunately, A(cid:98) satisfies (2) and (5) only by chance. The standardiza-
tion (5) can be achieved by modifications of logA(cid:98) that substract a suit-
able linear combination of logA(cid:98)(w) evaluated at certain values of w (cf. [3],
[17], [10]). In particular, [10] defines a nonparametric least squares estima-
ˆ ˆ
tor (nonparametric OLS) A(cid:98) (w) by logA(cid:98) (w) = β (w) where β (w)
OLS OLS 0 0
7
is obtained by least squares regression of logξ (w) − γ (i = 1,2,...,n) on
i
−logξ (e )−γ,...,−logξ (e )−γ. The resulting estimate satisfies (5), but
i 1 i d
it still may fail to satisfy (2).
In contrast, in the parametric approach introduced above a modification
is not needed, because the projection on A automatically leads to a proper
p
Pickands dependence function. However, the examples below demonstrate
thattheinitialestimatorremainscrucialforboththeasymptoticdistribution
and the finite sample behaviour of the parametric projection. Specifically,
we will compare the parametric approach based on A(cid:98)and A(cid:98) respectively.
OLS
Classical results on empirical processes yield a functional central limit
theorem of the following form (see [10] and references therein). Let C(∆ )
d
denote the Banach space of real-valued continuous functions on ∆ equipped
d
with the supremum norm. Then, as n → ∞,
√ (cid:16) (cid:17)
w
n A(cid:98)(w)−A(w) → A(w)ζ(w) =: ζ (w) in C(∆ ) (8)
A,nonp d
where ζ(w) (w ∈ ∆ ) is a zero mean Gaussian process with covariance
d
function
γ (v,w) := cov(ζ(v),ζ(w)) = cov(−logξ(v),−logξ(w)) (v,w ∈ ∆ ).
ζ d
Note that the joint distribution of ξ (v),ξ (w) does not depend on i, so that
i i
dropping the index i here does not lead to confusion. This result implies
in particular that, for large n and w ,...,w ∈ ∆ , the joint distribution
1 N d
of A(cid:98)(w ),...,A(cid:98)(w ) can be approximated by an N−dimensional normal
1 N
distribution with mean
µ(w ...,w ) = (A(w ),...,A(w ))T
1, N 1 N
and covariance matrix n−1Σ = n−1[σ(w ,w )] with
i j i,j=1,...,N
σ(w ,w ) = A(w )A(w )γ (w ,w )
i j i j ζ i j
= A(w )A(w )cov(−logξ(w ),−logξ(w )). (9)
i j i j
Now, givenaparametricclassofspectralmeasuresP basedonΨ ...,Ψ , an
p 1 p
estimator of the corresponding Pickands dependence function A(·,θ) ∈ A
p
can be defined as follows. Let A(cid:98) be the preliminary estimator in (7), and
denote by A (i = 1,...,p) the Pickands dependence functions correspond-
i
ing to the spectral measures Ψ (i = 1,...,p). Define a grid of w-values
i
8
w0,...,w0 ∈ ∆ , and the vector ˆa0 = (aˆ0,...,aˆ0 )T with aˆ0 = aˆ(w0)
1 N d 1 N i i
(i = 1,...,N) and
aˆ(w) = A(cid:98)(w)−A (w), w ∈ ∆ .
p d
Furthermore, define the N × (p−1) matrix H0 = [h (w0)]
j i i=1,...,N;j=1,...,p−1
with
h (w) = A (w)−A (w), w ∈ ∆ .
j j p d
Then the least squares estimator of θ is equal to
θˆ= Q0ˆa0
where Q0 = (H0TH0)−1H0T. Since the dependence functions A (w) are
j
defined for all w ∈ ∆ , an estimate of A(w) is available for any w ∈ ∆ by
d p
setting
A(w,θˆ) = A (w)+hT (w)θˆ= A (w)+hT (w)Q0ˆa0
p p
where h(w) = (h (w),...,h (w))T.
1 p−1
Letting N tend to infinity, an estimator of θ based on all values in ∆
d
can be obtained as follows. Suppose that the grid w0,...,w0 is chosen
1 N
such that, as N → ∞, the point measure M (B) = N−1(cid:80)N 1{w0 ∈ B}
N i=1 i
(B ∈ B(∆ )) converges weakly to a probability measure M on ∆ with
d d
Lebesgue density m(·). This can be achieved by deterministic or by random
choice (by sampling from M) of the grid. Then
N (cid:90)
N−1(cid:16)H0TH0(cid:17) = N−1(cid:88)h (w0)h (w0) = h (w)h (w)dM (w)
i l j l i j N
i,j ∆
l=1 d
converges to
(cid:90)
s = h (w)h (w)m(w)dw.
i,j i j
∆
d
This follows from the continuity of all A (and hence all h ) and from the
j j
compactness of ∆ . Similarly, the limit of
d
N
N−1(cid:16)H0Tˆa0(cid:17) = N−1(cid:88)h (w0)aˆ(w0)
j l l
j
l=1
9
is
(cid:90)
r = h (w)aˆ(w)m(w)dw.
j j
∆
d
Thus we obtain an estimator that depends on the density function m,
θˆ = S−1r
M
where S = (s ) and r = (r ,...,r )T. Even more generally, the
i,j i,j=1,...,p−1 1 p−1
previous estimators can be seen as special cases of
θˆ = S−1r (10)
M
where
(cid:90)
s = h (w)h (w)dM(w),
i,j i j
∆
d
(cid:90)
r = h (w)aˆ(w)dM(w)
j j
∆
d
and M is any distribution function on ∆ such that S is of full rank.
d
The same approach can be applied to any initial nonparametric estimator
of A for which a functional limit theorem is available. In particular, for the
nonparametric OLS, A(cid:98) , Gudendorf and Segers ([10]) obtain
OLS
√ (cid:16) (cid:17)
n A(cid:98) (w)−A(w) →w A(w)(cid:2)ζ(w)−λT (w)ζ(e)(cid:3)
OLS opt
=: ζ (w) in C(∆ )
A,OLS,nonp d
whereζ(w)istheGaussianprocessdefinedin(8),ζ(e) = (ζ(e ),...,ζ(e ))T,
1 d
(cid:2) (cid:3)
λ (w) = Σ−1E[ζ(e)ζ(w)]andΣ =E ζ(e)ζT (e) . Applyingtheparamet-
opt
ric approach, we define as before
θˆ = S−1r (11)
M,OLS OLS
with
(cid:90)
r = h (w)aˆ (w)dM(w)
OLS,j j OLS
∆
d
and
aˆ (w) = A(cid:98) (w)−A (w), w ∈ ∆ .
OLS OLS p d
10
