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Estimating asymptotic dependence functionals in
multivariate regularly varying models
Georg Mainik
∗
June 5, 2012
Abstract
This paper deals with semiparametric estimation of the asymptotic
portfolio risk factor γ
ξ
introduced by Mainik and R¨uschendorf (2010)
for multivariate regularly varying random vectors in R
d
+
. The func
tional γ
ξ
depends on the spectral measure Ψ, the tail index α, and
the vector ξ of portfolio weights. The representation of γ
ξ
is extended
to characterize the portfolio loss asymptotics for random vectors in
R
d
. The earlier results on uniform strong consistency and uniform
asymptotic normality of the estimates of γ
ξ
are extended to the gen
eral setting, and the regularity assumptions are signiﬁcantly weakened.
Uniform consistency and asymptotic normality are also proved for the
estimators of the functional γ
1/α
ξ
that characterizes the asymptotic be
haviour of the portfolio loss quantiles. The techniques developed here
can also be applied to other dependence functionals.
Keywords: tail dependence; multivariate regular variation; portfolio risk;
functional CLT; functional SLLN.
2010 Mathematics Subject Classiﬁcation: 62G32; 62H12; 60F17; 62P05
1 Introduction
Modelling dependence in multivariate random vectors and the estimation of
various dependence characteristics is of importance in many applications of
probability theory and statistics. In the areas of ﬁnance and insurance, any
portfolio modelling eﬀort includes dependence modelling and aggregation
of random variables with nontrivial dependence structures. Beyond under
standing portfolio behaviour under benign market conditions, the sensitivity
of the portfolio to extremal events deserves special attention. The aggrega
tion of risk is also related to risk diversiﬁcation, i.e., assessing the risk of a
weighted sum
P
d
i=1
ξ
(i)
X
(i)
for a random vector X = (X
(1)
, . . . , X
(d)
) and
portfolio weights ξ
(1)
, . . . , ξ
(d)
.
∗
RiskLab, Department of Mathematics, ETH Z¨urich, georg.mainik@math.ethz.ch
1
G. Mainik Estimating asymptotic dependence functionals
A useful mathematical framework for modelling the extremal behaviour
of random vectors is multivariate extreme value theory. Starting with the
characterization of joint distributions for componentwise maxima by de Haan
and Resnick (1977), multivariate extreme value theory found many appli
cations in insurance and ﬁnance (cf. McNeil et al., 2005; Malevergne and
Sornette, 2006). The literature on modelling and estimation of extremal
dependence is vast, including several alternative approaches, such as tail
dependence functions, spectral measures and alike, and extreme value cop
ulas. See Falk et al. (1994); de Haan and Ferreira (2006); Resnick (2007);
Gudendorf and Segers (2010), and references therein.
This paper presents a semiparametric approach to the estimation of tail
dependence functionals for multivariate regularly varying models. Particular
consequences of this assumption are that the loss components are heavy
tailed and that the extremal events for diﬀerent components are on the
same scale. This allows focusing on the cases with nontrivial contribution
of dependence to the extremal behaviour. The overall excess severity is
characterized by the tail index α := sup{β ≥ 0 : EkXk
β
< ∞}, which is the
number that separates the ﬁnite absolute moments from the inﬁnite ones.
In nondegenerate cases, all components X
(i)
have the same tail index
α. This computational advantage of multivariate regularly varying models
is also their most critical issue. It is well known that in case of diﬀerent
tail indices the asymptotic distribution of the portfolio loss is dominated
by the components with the heaviest tail, i.e., the one or the few X
(i)
with
the smallest α. One should always bear this potential issue in mind when
assuming multivariate regular variation. However, if the component tail
indices are close to each other, the excess behaviour over ﬁnite thresholds
may be better described by a model with equally heavy component tails.
Hence, from the practical point of view, multivariate regular variation can
be understood as a workable approximation for the case of statistically in
distinguishable component tail indices. Moreover, many popular models,
such as heavytailed elliptical or multivariate αstable distributions with
α ∈ (0, 2), are multivariate regularly varying (cf. Hult and Lindskog, 2002;
Araujo and Gin´e, 1980). Thus results obtained in this modelling framework
contribute to the general understanding of asymptotic dependence.
The dependence structure in multivariate regularly varying models is
characterized by the socalled spectral measure Ψ, which is the asymptotic
probability distribution of excess directions on a unit sphere. In this mod
elling framework, the tail dependence coeﬃcient and many other dependence
characteristics can be represented in terms of integrals with respect to the
spectral measure Ψ.
The estimation of dependence characteristics in the framework of ex
treme value theory or, more speciﬁcally, of multivariate regular variation,
is by now a vital research ﬁeld. The estimation of Ψ in the bivariate case
was studied by Einmahl et al. (1993). A purely nonparametric approach to
2
G. Mainik Estimating asymptotic dependence functionals
the estimation of dependence structures in the more general framework of
multivariate extreme value theory was proposed by Einmahl et al. (2001).
This setting was recently reconsidered for maximum likelihood estimation by
Einmahl and Segers (2009). A parametric approach to the estimation of the
spectral measure of heavytailed elliptical distributions has been considered
by Kl¨uppelberg et al. (2007). Nonparametric estimation of tail dependence
beyond the framework of extreme value theory has been studied by Schmidt
and Stadtm¨uller (2006).
The contribution of the present paper to the estimation of extremal
dependence is primarily focused on the asymptotic portfolio risk factor
γ
ξ
= γ
ξ
(Ψ, α), which characterizes the inﬂuence of the tail dependence struc
ture and the portfolio weights ξ
(1)
, . . . , ξ
(d)
on the extremal behaviour of the
portfolio risk (cf. Mainik and R¨uschendorf, 2010; Mainik and Embrechts,
2012). For related results we also refer to Embrechts et al. (2009a); Barbe
et al. (2006). The functional γ
ξ
was originally deﬁned for random vectors in
R
d
+
, which is appropriate for applications in insurance. The initial approach
is now extended to random vectors in R
d
, which allows accounting for losses
and gains simultaneously. This is particularly useful in ﬁnancial applications
or in any other problem area that involves modelling the compensation be
tween losses and gains.
The estimator bγ
ξ
considered here is semiparametric, combining an ap
propriate estimator of the tail index α ∈ (0, ∞) with an empirical estimator
of the spectral measure Ψ. The main results are the strong consistency
and the asymptotic normality of bγ
ξ
uniformly in the portfolio vector ξ. In
the special case of R
d
+
valued random vectors, a functional Law of Large
Numbers and a functional Central Limit Theorem for bγ
ξ
have been estab
lished by Mainik and R¨uschendorf (2010). The present paper extends these
results to the general case of random vectors in R
d
. Furthermore, the reg
ularity assumptions are relaxed. The statistical results are shown to hold
without any restrictions on the tail index α and the spectral measure Ψ.
Moreover, an explicit suﬃcient criterion is provided for the occurrence of
a nontrivial bias resulting from the estimation of the asymptotic angular
distribution. Finally, aiming at value at risk in ﬁnancial applications, this
paper includes the estimation of the functional γ
1/α
ξ
, which characterizes the
quantile asymptotics. The consistency and asymptotic normality results for
bγ
ξ
are extended to bγ
1/bα
ξ
.
Beyond extreme value theory, the proofs heavily rely on the theory of
empirical processes with functional index (cf. van der Vaart and Wellner,
1996). The plugin approach to the estimation of the functionals γ
ξ
and
γ
1/α
ξ
can be applied to other functionals of Ψ and α. (cf. Section 5). The
techniques in the proofs of the functional Laws of Large Numbers and the
functional Central Limit Theorems may be of wider interest.
The paper is structured as follows. Section 2 gives an introduction
3
G. Mainik Estimating asymptotic dependence functionals
to multivariate regular variation and the characterization of portfolio loss
asymptotics through the functional γ
ξ
. The estimation approach and the
main statistical results are presented in Section 3. The regularity assump
tions underlying the statistical results are discussed in Section 4. The con
clusions are drawn in Section 5, which also contains sketches of possible
generalizations. Section 6 contains the proofs of the results stated in this
paper.
2 Multivariate regular variation and asymptotic
dependence structures
Consider a random vector X = (X
(1)
, . . . , X
(d)
) in R
d
representing losses
and gains generated by some assets. Focusing on the risky side, let positive
component values X
(i)
represent losses and let the gains be indicated by
negative X
(i)
. Then the portfolio loss is given by
ξ
>
X :=
d
X
i=1
ξ
(i)
X
(i)
,
where ξ = (ξ
(1)
, . . . , ξ
(d)
) is a vector of portfolio weights.
As a special case, this notation includes relative losses of assets Z
(i)
in
a oneperiod model. Setting X
(i)
:= (Z
(i)
0
− Z
(i)
1
)/Z
(i)
0
, one obtains
ξ
>
X =
d
X
i=1
ξ
(i)
Z
(i)
0
Z
(i)
0
− Z
(i)
1
.
Thus ξ
>
X represents the random loss generated by investing the value ξ
(i)
in the ith asset, and the relative portfolio loss equals ξ
>
X/
P
d
i=1
ξ
(i)
.
Following the intuition of diversifying a unit amount of capital, let ξ be
restricted to a subset H of the hyperplane
H
1
:=
n
x ∈ R
d
: x
(1)
+ . . . + x
(d)
= 1
o
.
A particularly important special case is the exclusion of negative portfolio
weights (socalled short positions). The corresponding portfolio set is the
unit simplex:
Σ
d
:=
n
x ∈ R
d
+
: x
(1)
+ . . . + x
(d)
= 1
o
.
Aiming at dependence of extremes, we assume throughout the paper
that the probability distribution of X features a nontrivial dependence
structure in the tails. To introduce the necessary notions, let us start with
the deﬁnition of a regularly varying function.
4
G. Mainik Estimating asymptotic dependence functionals
A Lebesgue measurable function f : R
+
→ R
+
is called regularly varying
(at ∞) if there exists a function g : R
+
→ R
+
such that
∀x ∈ R
+
lim
t→∞
f(tx)
f(t)
= g(x).
It is well known that g is necessarily equal to x
β
for some β ∈ R. In case
β = 0 the function f is called slowly varying. Furthermore, regular variation
of f is equivalent to
f(x) = x
β
l(x)
with a slowly varying l. For further details on regular variation of functions
the reader is referred to the monograph by Bingham et al. (1987). We often
denote f ∈ RV
β
.
A random variable Y in R
+
is called regularly varying with tail index
α ≥ 0 if the corresponding tail probability
¯
F
Y
(x) := 1 − F
Y
(x) is regularly
varying with index −α:
¯
F
Y
∈ RV
−α
.
For brevity and convenience, the short notation Y ∈ RV
−α
will also be used
for regular variation of random variables. In case of random variables in R
regular variation can be considered separately for the lower and the upper
tail with an additional balance condition (cf. Resnick, 2007, Section 6.5.5).
The following deﬁnition provides the central model assumption of the
paper.
Deﬁnition 2.1. A random vector X in R
d
is multivariate regularly varying
if there exist a sequence a
n
→ ∞ and a (nonzero) Radon measure ν on
the Borel σﬁeld B([−∞, ∞]
d
\ {0}) such that ν([−∞, ∞]
d
\ R
d
) = 0 and,
as n → ∞,
nP
a
−1
n
X
v
→ ν on B
[−∞, ∞]
d
\ {0}
, (1)
where
v
→ denotes vague convergence of Radon measures and P
a
−1
n
X
is the
probability distribution of a
−1
n
X.
If X is restricted to R
d
+
, then ν is concentrated on [0, ∞]
d
\{0}. Therefore
multivariate regular variation in this special case can also be deﬁned by
vague convergence on B([0, ∞]
d
\{0}). For a full account of technical details
related to multivariate regular variation, vague convergence, and the Borel
σﬁelds on the punctured spaces [−∞, ∞]
d
\{0} and [0, ∞]
d
\{0} the reader
is referred to Resnick (2007).
It is well known that the limit measure ν obtained in (1) is unique
except for a constant factor, has a singularity in the origin in the sense that
ν((−ε, ε)
d
) = ∞ for any ε > 0, and exhibits the scaling property
ν(tA) = t
−α
ν(A) (2)
5
G. Mainik Estimating asymptotic dependence functionals
for some α > 0 and all sets A ∈ B
[−∞, ∞]
d
\ {0}
that are bounded away
from 0.
Furthermore, (1) implies that kXk ∈ RV
−α
for any norm k·k on R
d
. The
sequence a
n
can always be chosen as
a
n
:= F
←
kXk
(1 −1/n),
where F
←
kXk
is the quantile function of kXk. The resulting limit measure ν
is normalized by
ν
n
x ∈ R
d
: kxk > 1
o
= 1. (3)
In addition to (1) we assume that the limit measure ν is nondegenerate
in the following sense:
ν
n
x ∈ R
d
:
x
(i)
> 1
o
> 0, i = 1, . . . , d. (4)
This assumption ensures that all components X
(i)
are relevant for the ex
tremes of ξ
>
X. If (4) is satisﬁed in the upper tail region, i.e., if
ν
n
x ∈ R
d
: x
(i)
> 1
o
> 0, i = 1, . . . , d,
then ν also characterizes the asymptotic distribution of the componentwise
maxima M
n
:= (M
(1)
, . . . , M
(d)
) obtained from an i.i.d. sequence X
1
, . . . , X
n
via M
(i)
:= max{X
(i)
1
, . . . , X
(i)
n
}. In this case one has
a
−1
n
M
n
w
→ Y ∼ G
with the limit distribution function
G(y) := exp
−ν
[−∞, ∞]
d
\ [−∞, y]
, y ∈ (0, ∞]
d
.
Therefore ν is called exponent measure. For further details on the asymptotic
distributions of maxima see Resnick (1987) and de Haan and Ferreira (2006).
It is also well known that the scaling property (2) implies a product
representation of ν in polar coordinates with respect to any norm k·k on R
d
:
(r, s) := τ (x) := (kxk, kxk
−1
x).
The induced measure ν
τ
:= ν ◦ τ
−1
necessarily satisﬁes
ν
τ
= c ·ρ
α
⊗ Ψ (5)
with the constant factor c = ν({x ∈ R
d
: kxk > 1}), the measure ρ
α
on (0, ∞]
deﬁned by
ρ
α
((x, ∞]) := x
−α
, x ∈ (0, ∞],
6
G. Mainik Estimating asymptotic dependence functionals
and a probability measure Ψ on the unit sphere induced by k·k,
S
d
k·k
:=
n
s ∈ R
d
: ksk = 1
o
.
The measure Ψ is called spectral or angular measure of X. In the sequel
we normalize ν according to (3). This entails c = 1 in (5). Using this
normalization, it is easy to see that (1) is equivalent to
L
τ
t
−1
X
kXk > t
w
→ ρ
α
⊗ Ψ, t → ∞, (6)
on B((1, ∞] × S
d
k·k
). This suggests the notion of multivariate regular varia
tion with tail index α and spectral measure Ψ, abbreviated by X ∈ MRV
−α,Ψ
.
In the special case of R
d
+
valued random vectors X it may be convenient to
reduce the domain of Ψ to S
d
k·k
∩ R
d
+
.
Although the domain of Ψ depends on the norm underlying the polar
coordinates, the representation (5) is normindependent. If (5) holds for
some norm k·k, then it is also true for any other norm k·k
that is equivalent
to k·k. In the following we use the sum norm kxk
1
:=
P
d
i=1
x
(i)
 and let Ψ
denote the spectral measure on the unit sphere S
d
1
induced by k·k
1
.
Finally, multivariate regular variation of a random vector X is closely
related to the univariate regular variation of portfolio losses ξ
>
X. In non
degenerate cases, X ∈ MRV
−α,Ψ
implies that ξ
>
X ∈ RV
−α
for all ξ. For
further details and for the inverse, Cram´erWold type results we refer to
Basrak et al. (2002) and Boman and Lindskog (2009).
The property of multivariate regular variation appears in many popular
stochastic models. The examples include heavytailed elliptical distributions
(cf. Hult and Lindskog, 2002) and various copula models (cf. Alink et al.,
2004; Barbe et al., 2006; Embrechts et al., 2009b,a). Finally, (5) shows
that any combination of a probability measure Ψ on S
d
with a heavytailed
distribution on R
d
+
leads to a multivariate regular varying model. Deviation
from the exact product structure in the polar coordinates can be easily
implemented by distortions that disappear in the tail region.
More details on regular variation of functions or random variables can be
found in Bingham et al. (1987); Resnick (1987); Basrak et al. (2002); Hult
and Lindskog (2006); de Haan and Ferreira (2006); Resnick (2007).
The following theorem provides a characterization of the asymptotic
portfolio losses in multivariate regularly varying models. The special case of
random vectors in R
d
+
was studied in Mainik and R¨uschendorf (2010). The
asymptotic portfolio risk factor γ
ξ
introduced there and called extreme risk
index of the portfolio ξ has an immediate generalization for random vectors
in R
d
.
Lemma 2.2. Let X ∈ MRV
−α,Ψ
, α > 0. Then
(a)
lim
t→∞
P{ξ
>
X > t}
P{kXk
1
> t}
= γ
ξ
:=
Z
S
d
1
(ξ
>
s)
α
+
dΨ(s); (7)
7
G. Mainik Estimating asymptotic dependence functionals
(b)
lim
u↑1
F
←
ξ
>
X
(u)
F
←
kXk
1
(u)
= γ
1/α
ξ
. (8)
Proof. See Mainik and R¨uschendorf (2010), equations (3.1) and (2.8).
The immediate consequence of (7) and (8) is that the functional γ
ξ
characterizes the asymptotics of portfolio loss probabilities and the corre
sponding high loss quantiles. The limit relation (8) allows for an asymptotic
comparison of the value at risk associated with diﬀerent portfolio vectors ξ.
The value at risk VaR
1−λ
(Y ) of a random loss Y at the level 1 −λ is deﬁned
as the 1 −λ quantile of Y (cf. McNeil et al., 2005):
VaR
1−λ
(Y ) := F
←
Y
(1 −λ).
Further extensions to the asymptotic ordering of the expected shortfall ES
1−λ
and other spectral risk measures are also possible (cf. Mainik and R¨uschendorf,
2010).
3 Estimation
According to (7), the functional γ
ξ
is obtained by indexing the measure Ψ
by a function f
ξ,α
(s) := (ξ
>
x)
α
+
:
γ
ξ
= Ψf
ξ,α
:=
Z
f
ξ,α
(s) dΨ(s).
Combining an estimator
b
Ψ with an estimator bα, we obtain a plugin estima
tor for γ
ξ
:
bγ
ξ
:=
b
Ψf
ξ,bα
. (9)
A natural estimator for the functional γ
1/α
ξ
obtained in (8) would be bγ
1/bα
ξ
.
In the following we consider the estimation of γ
ξ
and γ
1/α
ξ
from an i.i.d.
sample X
1
. . . , X
n
∼ X. Working with polar coordinates of X in 1norm,
we will denote them by (R, S):
R := kXk
1
, S := kXk
−1
1
X.
Accordingly, R
i
and S
i
will denote the radial and angular parts of X
i
for
i = 1, . . . , n. All estimators are based upon the subsample related to k upper
order statistics R
n:1
, . . . , R
n:k
of the radial parts. The number k = k(n)
satisﬁes
k = k(n) → ∞, k/n → 0 as n → ∞.
To avoid technicalities, we assume that the distribution F
R
of the radial
parts is continuous:
F
R
∈ C(R
+
).
8
G. Mainik Estimating asymptotic dependence functionals
Then the sample indices i(n, 1) < . . . < i(n, k) of R
n:1
, . . . , R
n:k
are well
deﬁned almost surely and the corresponding angular parts can be written
as S
i(n,1)
, . . . , S
i(n,k)
. The resulting empirical estimator of Ψ is given by
b
Ψ = P
n
:=
1
k
k
X
j=1
δ
S
i(n,j)
. (10)
The estimator of the tail index α is supposed to be a function of R
n:1
, . . . , R
n:k
.
Various estimation approaches are possible (cf., among others, Hill, 1975;
Pickands, 1975; Smith, 1987; Dekkers et al., 1989). Instead of specifying the
tail index estimator bα, we will only impose assumptions on bα, such as strong
consistency or asymptotic normality. This allows to choose bα according to
the application.
The result cited below gives insight into the distribution structure of the
extreme subsample X
i(n,1)
, . . . , X
i(n,k)
and the corresponding angular parts
S
i(n,1)
, . . . , S
i(n,k)
. The proof is given in Mainik and R¨uschendorf (2010).
Lemma 3.1. Let X ∈ MRV
−α,Ψ
, assume F
R
∈ C(R
+
), and denote
U
n
:= F
R
(R
n:k+1
) .
Then, for any u ∈ (0, 1),
L
X
i(n,1)
, . . . , X
i(n,k)
U
n
= u
= ⊗
k
i=1
L(XF
R
(R) > u). (11)
An immediate consequence of (11) is
L
S
i(n,1)
, . . . , S
i(n,k)
U
n
= u
= ⊗
k
i=1
Ψ
u
, (12)
where
Ψ
u
:= L(SF
R
(R) > u) .
The conditional i.i.d. structure obtained in (12) can also be written as
P
S
i(n,1)
, . . . , S
i(n,k)
∈ A
=
Z
[0,1]
Ψ
k
u
(A) dP
U
n
(u).
Here A is a Borel subset of
S
d
1
k
, P
U
n
is the probability distribution of U
n
,
and Ψ
k
u
:= ⊗
k
i=1
Ψ
u
for u ∈ (0, 1). Moreover, since F
←
R
(u) → ∞ for u ↑ 1,
multivariate regular variation of X implies
Ψ
u
w
→ Ψ, u ↑ 1. (13)
The central results of the present paper are the uniform strong consis
tency and the uniform asymptotic normality of bγ
ξ
and bγ
1/bα
ξ
. These proper
ties are related to the theory of empirical measures and empirical processes.
Interested readers are referred to the monograph by van der Vaart and Well
ner (1996).
We start with strong consistency.
9
G. Mainik Estimating asymptotic dependence functionals
Theorem 3.2. Suppose that H ⊂ H
1
is compact, k(n) ≥ δn
q
for some
q ∈ (0, 1) and δ > 0, and bα is strongly consistent:
bα
a.s.
→ α.
Then bγ
ξ
is strongly consistent uniformly in ξ ∈ H:
sup
ξ∈H
bγ
ξ
− γ
ξ

a.s.
→ 0. (14)
Remark 3.3. According to Mason (1982), the n
q
growth rate for k(n) is
a natural assumption for the strong consistency of the Hill estimator bα
H
.
Since bα
H
is the most prototypical estimator for α, this assumption does not
restrict the applicability of Theorem 3.2.
Given (14), the uniform strong consistency of bγ
1/bα
ξ
follows from bα
a.s.
→
α ∈ (0, ∞) and the uniform continuity of the mapping (t, α) 7→ t
1/α
on
[0, K] × [ε, 1/ε] for any K, ε ∈ (0, ∞).
Corollary 3.4. The assumptions of Theorem 3.2 also imply that
sup
ξ∈H
bγ
1/bα
ξ
− γ
1/α
ξ
a.s.
→ 0.
For asymptotic normality, we need some regularity assumptions and ad
ditional notation.
Condition 3.5.
(a) bα is asymptotically normal:
√
k (bα − α)
w
→ Y ∼ N(µ
α
, σ
2
α
) (15)
(b) The random variable Y
n
:=
√
k(bα − α) and the mapping G
n
: Ω →
l
∞
(F
H,α
) deﬁned by
G
n
:=
√
k (P
n
− Ψ
U
n
) (16)
with the random centring
Ψ
U
n
(ω) := Ψ
U
n
(ω)
are asymptotically independent. That is, for any bounded continuous
functions h
1
∈ C
b
(R) and h
2
∈ C
b
(C(F
H,α
)) we have
lim
n→∞
E [h
1
(Y
n
)h
2
(G
n
)] − Eh
1
(Y
n
)Eh
2
(G
n
) = 0.
10
G. Mainik Estimating asymptotic dependence functionals
(c) There exists a mapping b ∈ l
∞
(H) such that
sup
ξ∈H
√
k (Ψ
U
n
− Ψ) f
ξ,α
− b(ξ)
P
→ 0. (17)
Remark 3.6. (a) Many popular estimators of the tail index α are asymptot
ically normal under appropriate secondorder conditions specifying the
convergence rate of the distribution L(t
−1
RR > t) for t → ∞. A com
prehensive elaboration on this topic can be found in de Haan and Fer
reira (2006). For original results see, among others, Davis and Resnick
(1984); Drees (1995); Dekkers et al. (1989); Smith (1987); Drees et al.
(2004).
(b) Condition 3.5(b) allows to leave bα in Theorem 3.7 unspeciﬁed. How
ever, since asymptotic independence of radial and angular parts is an
essential feature of multivariate regularly varying models, this assump
tion meets the natural intuition towards any sensible estimator bα =
bα(R
i(n,1)
, . . . , R
i(n,k)
) and the empirical process G
n
, constructed from
the angular parts S
i(n,1)
, . . . , S
i(n,k)
. In particular, the Hill estimator
bα
H
, representing one of the most fundamental approaches to the esti
mation of the tail index α, satisﬁes Condition 3.5(b) automatically. See
Lemma 4.3 and Corollary 4.4 for further details.
(c) Condition 3.5(c) can be understood as a secondorder condition for
the angular parts S
i(n,1)
, . . . , S
i(n,k)
. Since multivariate regular varia
tion leaves convergence rates completely unspeciﬁed, similar conditions
are necessary for establishing asymptotic normality in regularly varying
models. An explicit suﬃcient criterion for (17) is obtained in Lemma 4.1
and illustrated in Example 4.2.
In the following, let ∂
(·)
denote the partial derivative, e.g.,
∂
α
f
ξ,α
:=
∂
∂α
f
ξ,α
.
Further, let G
Ψ
denote the ΨBrownian bridge on a function class F, which
is a tight stochastic process with index f ∈ F and multivariate Gaussian
ﬁnitedimensional marginal distributions
(G
Ψ
f
1
, . . . , G
Ψ
f
m
) ∼ N(0, C). (18)
The covariance structure is determined by Ψ as follows:
C
i,j
= Ψ
f
i
− Ψf
i
f
j
− Ψf
j
= Ψf
i
f
j
− Ψf
i
Ψf
j
. (19)
The tightness of G
Ψ
implies that this process has a version with σ
Ψ
continuous
paths (cf. van der Vaart and Wellner, 1996, Section 2.1.2). The variance
seminorm σ
Ψ
is deﬁned by
σ
Ψ
(f) :=
Ψ
(f − Ψf)
2
1/2
. (20)
11
G. Mainik Estimating asymptotic dependence functionals
Now we can state the uniform asymptotic normality of bγ
ξ
.
Theorem 3.7.
(a) Let X ∈ MRV
−α,Ψ
, α > 0, and assume that Condition 3.5 is satisﬁed.
Then bγ
ξ
is asymptotically normal uniformly in ξ ⊂ H for compact H ⊂
H
1
:
√
k (bγ
ξ
− γ
ξ
)
w
→ b(ξ) + G
Ψ
f
ξ,α
+ Ψ [∂
α
f
ξ,α
] Y in l
∞
(H). (21)
Here, b(ξ) is the asymptotic bias term from (17), G
Ψ
is a ΨBrownian
bridge on F
H,α
, and Y is the Gaussian limit in (15). Thus Y is inde
pendent of G
Ψ
.
(b) Suppose that the assumptions of Part (a) are satisﬁed except for Condi
tion 3.5(c), which is satisﬁed only pointwise:
√
k(Ψ
U
n
f
ξ
i
,α
− Ψf
ξ
i
,α
) → b(ξ
i
) ∈ R (22)
for ξ
1
, . . . , ξ
p
∈ H. Then
√
k
bγ
ξ
1
, . . . , bγ
ξ
p
−
γ
ξ
1
, . . . , γ
ξ
p
w
→ N (M, C) (23)
where the mean vector M = M(α, ξ
1
, . . . , ξ
p
) and the covariance matrix
C = C(α, ξ
1
, . . . , ξ
p
) are given by
M
(i)
= b(ξ
i
) + µ
α
Ψ [∂
α
f
ξ
i
,α
] , (24)
C
i,j
= Ψ
f
ξ
i
,α
f
ξ
j
,α
− Ψf
ξ
i
,α
Ψf
ξ
j
,α
+ σ
2
α
Ψ [∂
α
f
ξ
i
,α
] Ψ
∂
α
f
ξ
j
,α
(25)
with i, j ranging in {1, . . . , p} and µ
α
, σ
2
α
from (15).
The asymptotic normality of bγ
ξ
extends to bγ
1/bα
ξ
.
Theorem 3.8. Suppose that H is compact and γ
ξ
6= 0 for all ξ ∈ H. Then
(a) The assumptions of Theorem 3.7(a) imply that
√
k
bγ
1/bα
ξ
− γ
1/α
ξ
w
→ c
1
Y + c
2
Z in l
∞
(H),
where Y is the Gaussian limit in (15) and Z is the right side of (21).
The constant factors c
i
are given by
c
1
:= −
1
α
2
γ
1/α
ξ
log γ
ξ
and c
2
:=
1
α
γ
1/α−1
ξ
.
(b) The assumptions of Theorem 3.7(b) imply that
√
k
bγ
1/bα
ξ
1
, . . . , bγ
1/bα
ξ
p
−
γ
1/α
ξ
1
, . . . , γ
1/α
ξ
p
w
→ N
c
1
µ
α
+ c
2
M, c
2
1
σ
2
α
+ c
2
2
C
with M and C deﬁned in (24) and (25), respectively.
12
G. Mainik Estimating asymptotic dependence functionals
4 Examples and comments
This section is dedicated to the regularity assumptions underlying Theo
rem 3.7. We start with an explicit criterion that implies Condition 3.5(c).
Lemma 4.1. Let X ∈ MRV
−α,Ψ
, H ⊂ H
1
compact, and F
R
∈ C(R
+
).
Suppose that
√
k
Ψ
1−k/n
− Ψ
f
ξ,α
→ b(ξ) in l
∞
(H) (26)
and that the mapping u 7→ Ψ
0
u
f
ξ,α
with Ψ
0
u
:= L(SF
R
(R) = u) is continuous
in u ∈ (0, 1] for any ξ ∈ H. Then
√
k (Ψ
U
n
− Ψ) f
ξ,α
P
→ b(ξ) in l
∞
(H).
The following example illustrates Lemma 4.1 and shows that the angular
bias term b = b(ξ) depends on the choice of the extreme subsample size
k = k(n).
Example 4.2. Consider a multivariate regularly varying distribution with
conditional angular distribution Ψ
0
u
:= L(SF
R
(R) = u) given by
Ψ
0
u
:= uΨ
0
1
+ (1 − u)Ψ
0
0
,
where Ψ
0
1
and Ψ
0
0
are arbitrary probability measures on B(S
d
1
). Given the
continuity of the radial distribution F
R
, the conditional angular distribution
Ψ
u
:= L(SF
R
(R) > u) is equal to
Ψ
u
=
1
1 − u
Z
(u,1)
Ψ
0
v
dv
= Ψ
0
1
1
1 − u
Z
(u,1)
v dv + Ψ
0
0
1
1 − u
Z
(u,1)
(1 − v) dv
=
1 + u
2
Ψ
0
1
+
1 − u
2
Ψ
0
0
.
This yields that the spectral measure Ψ is equal to Ψ
0
1
, and therefore
Ψ
1−k/n
− Ψ =
−k/n
2
Ψ
0
1
+
k/n
2
Ψ
0
0
=
k
2n
Ψ
0
0
− Ψ
0
1
.
Hence condition (26) is equivalent to
k
3/2
2n
Ψ
0
0
− Ψ
0
1
f
ξ,α
→ b(ξ) in l
∞
(H).
Consequently, (26) is satisﬁed if k
3/2
/(2n) → λ ∈ [0, ∞). The asymptotic
bias term b(ξ) appearing in Theorem 3.7 is given by
b(ξ) = λ
Ψ
0
0
− Ψ
0
1
f
ξ,α
.
In particular, b(ξ) is nonzero for λ > 0.
13
G. Mainik Estimating asymptotic dependence functionals
Another point that is worth a discussion is the asymptotic independence
of the normalized estimation error Y
n
=
√
k(bα − α) and the empirical pro
cess G
n
stated in Condition 3.5(b). As already highlighted in Remark 3.6,
this condition is rather natural in the framework of multivariate regular
variation and is automatically satisﬁed by the Hill estimator. The rest of
the current section provides a proof for this assertion.
The Hill estimator (cf. Hill, 1975), deﬁned as
bα
H
:=
1
k
k
X
i=1
log(R
n:i
/R
n:k+1
)
!
−1
,
is one of the earliest and most popular estimators for the tail index α of
a heavytailed distribution. Denoting
e
R
i(n,j)
:= R
i(n,j)
/R
n:k+1
, one obtains
the representation
bα
−1
H
=
1
k
k
X
j=1
log
e
R
i(n,j)
.
Hence the tuple (bα
−1
H
, P
n
f
ξ,α
) can be written as
bα
−1
H
, P
n
f
ξ,α
=
e
P
n
˜
l,
e
P
n
˜
f
ξ,α
(27)
with the empirical measure
e
P
n
deﬁned by
e
P
n
:=
1
k
k
X
i=1
δ
(
e
R
i(n,j)
,S
i(n,j)
)
and the functional indices
˜
l,
˜
f
ξ,α
deﬁned by
˜
l(r, s) := log(r) and
˜
f
ξ,α
(r, s) := f
ξ,α
(s).
Recall that R
n:k+1
= F
←
R
(U
n
) Pa.s. for continuous F
R
. Consequently,
Lemma 3.1 yields
L
e
R
i(n,1)
, S
i(n,1)
, . . . ,
e
R
i(n,k)
, S
i(n,k)
U
n
= u
= ⊗
k
i=1
e
P
u
where
e
P
u
:= L(R/F
←
R
(u), SF
R
(R) > u) . (28)
The representation (27) shows that the asymptotic independence of the
normalized estimation error Y
n
:=
√
k(bα
H
− α) and the empirical process
G
n
assumed in Condition 3.5(b) is related to the asymptotic behaviour of
the empirical process
e
G
n
:=
√
k
e
P
n
−
e
P
U
n
(29)
14
G. Mainik Estimating asymptotic dependence functionals
with the random centring
e
P
U
n
(ω) :=
e
P
U
n
(ω)
and functional index
f ∈
e
F
H,α
:=
n
˜
l
o
∪
n
˜
f
ξ,α
: ξ ∈ H
o
.
The following lemma states weak convergence of the empirical process
e
G
n
to a Gaussian process.
Lemma 4.3. Let X ∈ MRV
−α,Ψ
, α > 0, H ⊂ H
1
compact, and F
R
∈
C(R
+
). Then the empirical process
e
G
n
deﬁned in (29) satisﬁes
e
G
n
w
→ G
ρ
α
⊗Ψ
in C
e
F
H,α
. (30)
The ﬁnal result of this section is obtained by combination of Lemma 4.3
with the DeltaMethod.
Corollary 4.4. Suppose that the conditions of Lemma 4.3 are satisﬁed and
√
k
e
P
U
n
˜
l − α
−1
P
→ b ∈ R. (31)
Then bα
H
is asymptotically normal and satisﬁes Condition 3.5(b), i.e., the
random variable Y
n
:=
√
k(bα
H
− α) is asymptotically independent from G
n
.
Remark 4.5. It is well known that condition (31) can be ensured by strength
ening the regular variation of the radial part R by a secondorder condition
and adding a regularity condition on the sequence k = k(n). For techni
cal details on the asymptotic normality of tail index estimates we refer to
de Haan and Ferreira (2006).
5 Conclusions and generalizations
The approach to the comparison of extremal portfolio losses proposed by
Mainik and R¨uschendorf (2010) for multivariate regularly varying vectors
in R
d
+
has been extended to R
d
. The statistical results from Mainik and
R¨uschendorf (2010) have also been extended to this more general case. Anal
ogous results have been obtained for the estimator bγ
1/bα
ξ
.
Moreover, the conditions underlying the asymptotic normality results
have been signiﬁcantly relaxed by dropping all restrictions on α and Ψ.
A thorough discussion of these conditions has been provided, including a
suﬃcient criterion for the appearance of the nontrivial angular bias b(ξ).
If the model has exact product structure, then the angular bias is zero. In
particular, this is the case for centred multivariate tdistributions. Models
featuring asymptotically vanishing distortion of the product structure may
have nontrivial angular bias, depending on the decay rate of the distortion.
15
G. Mainik Estimating asymptotic dependence functionals
The combination of extreme value theory with the theory of empirical
processes presented here is also applicable to other problems in the area of
extremal dependence. For instance, instead of the portfolio excess sets
n
x ∈ R
d
+
: ξ
>
x > t
o
one may be interested in the asymptotic probabilities of the sets
x ∈ R
d
+
: ∧
i
ξ
(i)
x
(i)
> t
with direction parameter ξ ∈ H ⊂ Σ
d
. These sets indicate that at least one
of the weighted components ξ
(i)
X
(i)
exceeds t. Analogously to Lemma 2.2
one obtains that the asymptotic probabilities of these sets for t → ∞ can
be quantiﬁed by the functional Ψg
ξ,α
with g
ξ,α
(s) = (∨
i
ξ
(i)
s
(i)
)
α
. Similarly
to (8), the asymptotic dependence factor for the corresponding quantiles is
equal to (Ψg
ξ,α
)
1/α
.
If asymptotic independence is excluded, same arguments can be ap
plied to the asymptotic probabilities of the directed joint excess events
{x ∈ R
d
+
: ∧
i
ξ
(i)
x
(i)
> t}. This leads to similar limit functionals with the
integrand g
ξ,α
(s) = (∧
i
ξ
(i)
s
(i)
)
α
.
Combining a semiparametric estimation approach and the techniques of
the proof presented in Section 6, it is straightforward to derive functional
Laws of Large Numbers and functional Central Limit Theorems similar to
the Theorems 3.2, 3.7, 3.8, and Corollary 3.4.
6 Proofs
6.1 Empirical processes with functional index
The estimator bγ
ξ
proposed in (9) is obtained by indexing the empirical
measure P
n
deﬁned in (10) with a random element f
ξ,bα
of the function class
F
H
:= {f
ξ,α
: ξ ∈ H, α ∈ (0, ∞)},
where H ⊂ H
1
is a compact set of admissible portfolio vectors. Later
on we will see that consistency of the estimator bα and smoothness of the
parametrization α 7→ f
ξ,α
allow to reduce the index set of the empirical
measure P
n
and the empirical process G
n
deﬁned in (16) to the function
class
F
H,α
:= {f
ξ,α
: ξ ∈ H}
with α ∈ (0, ∞) being the true tail index.
The asymptotic normality of bγ
ξ
can be viewed as a special version of the
Donsker Theorem. Let P
k,Ψ
denote the empirical measure corresponding to
16
G. Mainik Estimating asymptotic dependence functionals
k i.i.d. random variables with probability distribution Ψ:
P
k,Ψ
:=
1
k
k
X
i=1
δ
Y
i
, Y
1
, . . . , Y
k
i.i.d. ∼ Ψ. (32)
The corresponding empirical process G
k,Ψ
is deﬁned as
G
k,Ψ
:=
√
k (P
k,Ψ
− Ψ) .
A class F of measurable functions is called preGaussian if there exists a
tight ΨBrownian bridge G
Ψ
on F. F is called Donsker if the Donsker
Theorem holds for G
k,Ψ
uniformly in f ∈ F:
G
k,Ψ
w
→ G
Ψ
in l
∞
(F), k → ∞. (33)
The preGaussian and Donsker properties of F guarantee the existence of
a probability space (Ω
0
, A
0
, P
0
) and a tight, Borel measurable mapping G
Ψ
:
Ω
0
→ l
∞
(F) satisfying (18), (19), and (33).
The notion of weak convergence in l
∞
(F) is understood according to
van der Vaart and Wellner (1996). Based on outer expectations and outer
probabilities, this extended notion allows to consider P
n
and G
n
as mappings
from the probability space Ω into l
∞
(F), although the measurability in l
∞
is not available in general (cf. Billingsley, 1968, Section 18). In the special
case when P
n
and G
n
are measurable, the extended notions of stochastic
convergence (weak, in probability, or almost sure) coincide with the standard
ones. As it will be shown below, the problem considered here is of this kind.
This allows to apply Donsker Theorems from van der Vaart and Wellner
(1996) and obtain weak convergence of empirical measures in the classical
sense.
However, standard Donsker Theorems for i.i.d. samples cannot be ap
plied to the subsample S
i(n,1)
, . . . , S
i(n,k)
directly. Although conditionally
i.i.d. given U
n
= u (cf. Lemma 3.1), the random variables S
i(n,1)
, . . . , S
i(n,k)
are not necessarily independent. Moreover, the probability distribution of
each S
i(n,j)
varies with n. Thus uniform convergence results for and G
n
f
ξ,bα
demand a special version of the Donsker Theorem that takes into account
the structure of the underlying probability distribution. This result is stated
in Lemma 6.5, after a series of auxiliary results.
We start with an outline of some useful facts.
Remark 6.1. (a) The mapping (ξ, s, α) → f
ξ,α
(s) is continuous, and hence
uniformly continuous on a compact domain. This implies that any func
tion class F
H,I
:= {f
ξ,α
: ξ ∈ H, α ∈ I} with compact H and compact
I ⊂ (0, ∞) is uniformly bounded. Moreover, such F
H,I
are compact
in (C(S
d
1
), k·k
∞
). In particular, this is the case for all function classes
F
H,α
:= {f
ξ,α
: ξ ∈ H} with compact H. Same arguments apply to the
partial derivatives ∂
α
f
ξ,α
and ∂
2
α
f
ξ,α
and the corresponding function
classes ∂
α
F
H,I
and ∂
2
α
F
H,I
.
17
G. Mainik Estimating asymptotic dependence functionals
(b) It is obvious that any probability measure Ψ on S
d
1
satisﬁes Ψf − Ψg ≤
kf − gk
∞
for f, g ∈ C(S
d
1
). Thus the mapping f 7→ Ψf is Lipschitz with
factor 1.
Recall the conditional angular distribution Ψ
u
= L(SF
R
(R) > u) de
ﬁned for u ∈ [0, 1) (cf. Lemma 3.1). Motivated by (13), we introduce the
extended notation Ψ
1
:= Ψ. The subsequent lemma provides the continuity
of the parametrization u 7→ Ψ
u
, which is essential to the measurability of
the random centring Ψ
U
n
in (16).
Lemma 6.2. Let X ∈ MRV
−α,Ψ
with α ∈ (0, ∞), F
R
continuous, and
H ⊂ H
1
compact. Then
(a) the mappings u 7→ Ψ
u
f
ξ,α
and u 7→ Ψ
u
[∂
α
f
ξ,α
] are continuous in u ∈
[0, 1] for any ξ ∈ H;
(b) The measure Ψ
u
converges to Ψ in l
∞
:
kΨ
u
− Ψk
F
∗
:= sup
f∈F
∗
Ψ
u
f − Ψf → 0, u ↑ 1,
for F
∗
= F
H,α
and F
∗
= ∂
α
F
H,α
:= {∂
α
f
ξ,α
: ξ ∈ H}.
Proof. Part (a). Continuity of F
R
implies that F
R
(R) ∼ unif(0, 1), and
therefore
Ψ
u
f =
E [f(S)1{F
R
(R) > u}]
1 − u
for u < 1. Moreover, P{F
R
(R) = u} = 0 implies that
1{F
R
(R) > u
n
}
a.s.
→ 1{F
R
(R) > u}
for any sequence u
n
→ u < 1. Thus Ψ
u
n
f → Ψ
u
f follows from the Dom
inated Convergence Theorem. The continuity of u 7→ Ψ
u
f in u = 1 is an
immediate consequence of the weak convergence Ψ
u
w
→ Ψ = Ψ
1
established
in (13).
Part (b). According to Remark 6.1(b), the mapping f 7→ Ψf is Lip
schitz(1) for all Ψ. Hence the family {Ψ
u
: u ∈ [0, 1]} can be considered as
an equicontinuous subset of C(F
∗
). The uniform convergence Ψ
u
n
f → Ψ
u
f
for f ∈ F
∗
follows from the compactness of F
∗
stated in Remark 6.1(b) and
the pointwise convergence in Part (a).
The following result guarantees that the random measures involved in
the proof of Theorem 3.7 can be treated as random variables in C(F
∗
).
Lemma 6.3. The empirical measures P
n
and P
k,Ψ
, the random measures
Ψ
U
n
and the empirical processes G
n
and G
k,Ψ
are Borel measurable mappings
in C(F
∗
) for F
∗
= F
H,α
and F
∗
= ∂
α
F
H,α
.
18
G. Mainik Estimating asymptotic dependence functionals
Proof. According to Remark 6.1(a), F
∗
is a compact subset of C(S
d
1
). More
over, Remark 6.1(b) implies that the mappings f 7→ P
n
(ω)f, f 7→ P
k,Ψ
(ω)f,
f 7→ Ψ
U
n
(ω)f, f 7→ G
n
(ω)f, and f 7→ G
k,Ψ
(ω)f are continuous in f ∈ F
∗
for any ω ∈ Ω. Since F
∗
is compact, a mapping ω 7→ φ(ω) from Ω into
C(F
∗
) is Borel measurable if ω 7→ (φ(ω))(f) is measurable for all f ∈ F
∗
(cf.
van der Vaart and Wellner, 1996, Example 1.5.1). Thus it suﬃces to show
the measurability of the random variables P
n
f, P
k,Ψ
f and Ψ
U
n
f for every
f ∈ F
∗
. The measurability of G
n
f, and G
k,Ψ
f is a trivial consequence.
It is easy to see that the mappings ω 7→ P
n
(ω)f and ω 7→ P
k,Ψ
(ω)f for
f ∈ F
∗
are measurable by construction (cf. (10) and (32)). The measur
ability of ω 7→ Ψ
U
n
(ω)f = Ψ
U
n
(ω)
f follows immediately from the measur
ability of U
n
and the continuity of the mapping u 7→ Ψ
u
f established in
Lemma 6.2(a).
A function class F is called universally Donsker or preGaussian, if the
corresponding property holds uniformly for all probability measures on the
sample space.
Lemma 6.4. Let H ⊂ H
1
be compact and α ∈ (0, ∞). Then the function
class F
H,α
is universally Donsker and preGaussian.
Proof. It is obvious that all functions f ∈ F
H,α
are measurable and uni
formly bounded (cf. Remark 6.1(a)). Hence the constant function
F (s) := 1
S
d
1
(s) sup
f∈F
H,α
kfk
∞
can serve as an envelope function for F
H,α
. That is, we have F (s) ≥ f (s)
for all s ∈ S
d
1
and all f ∈ F
H,α
.
The separability of F
H,α
in l
∞
(S
d
1
) implies that F
H,α
is universally Ψ
measurable, i.e., Ψmeasurable for any probability measure Ψ on B(S
d
1
)
(cf. van der Vaart and Wellner, 1996, Deﬁnition 2.3.3). Another conse
quence of the separability of F
H,α
is the separability of the function classes
{f − g : f, g ∈ F
H,α
, kf − gk
Ψ,2
< δ} and {(f − g)
2
: f, g ∈ F
H,α
}. There
fore these function classes are universally Ψmeasurable, and Theorem 2.8.3
from van der Vaart and Wellner (1996) yields that F
H,α
is universally
Donsker and preGaussian if the following uniform entropy condition is sat
isﬁed:
Z
∞
0
sup
Q∈Q
q
log N(εkF k
Q,2
, F
H,α
, L
2
(Q)) dε < ∞. (34)
Here Q denotes the set of all ﬁnitely discrete probability measures, and
N(ε, F
H,α
, L
2
(Q)) is the number of εballs in L
2
(Q) needed to cover F
H,α
.
An εball around f is deﬁned as {g ∈ L
2
(Q) : kg − fk
Q,2
< ε}.
Notice that F
H,α
is covered by a singe ball of size kF k
Q,2
:
∀f ∈ F
H,α
kfk
Q,2
≤ kF k
Q,2
.
19
G. Mainik Estimating asymptotic dependence functionals
This allows to reduce (34) to
Z
1
0
sup
Q∈Q
q
log N(εkF k
Q,2
, F
H,α
, L
2
(Q)) dε < ∞. (35)
It is obvious that (ξ
>
1
s)
+
− (ξ
>
2
s)
+
 ≤ kξ
1
− ξ
2
k
∞
for s ∈ S
d
1
. This implies
that the mapping φ
α
: ξ 7→ f
ξ,α
from H to C(S
d
1
) is Lipschitz for α ≥ 1,
whilst for α ∈ (0, 1) we obtain that
kφ
α
(ξ
1
) − φ
α
(ξ
2
)k
∞
= sup
s∈S
d
1
ξ
>
1
s
α
+
−
ξ
>
2
s
α
+
≤ sup
s∈S
d
1
ξ
>
1
s
+
−
ξ
>
2
s
+
α
≤ kξ
1
− ξ
2
k
α
.
Hence, if we cover H by m balls of radius δ, the set F
H,α
= φ
α
(H) is
covered by m balls of radius cδ
α∧1
(in k·k
∞
) for some c > 0. Being a
compact subset of R
d
, H can be covered by m = O(δ
−d
) balls of radius δ.
Therefore F
H,α
can be covered by O(ε
−d/(α∧1)
) balls of size ε. As any ball
in k·k
∞
is smaller than the ball in any L
2
(Q) metric with the same center
and radius, the polynomial bound O(ε
−d/(α∧1)
) also holds for the covering
number N in (35) uniformly in Q. Hence the integrability condition (35) is
satisﬁed.
Now we can prove the weak convergence of the empirical process G
n
deﬁned in (16).
Lemma 6.5. Suppose that X ∈ MRV
−α,Ψ
and H ⊂ H
1
is compact. Then
the empirical process G
n
=
√
k(P
n
− Ψ
U
n
) satisﬁes
G
n
w
→ G
Ψ
in C(F
H,α
).
Proof. According to Lemma 6.3, G
n
is a Borel measurable mapping into
C(F
H,α
). Thus weak convergence is understood in the classical way, and it
suﬃces to show that
lim
n→∞
Eh(G
n
) = Eh(G
Ψ
) (36)
for any function h ∈ C
b
(C(F
H,α
)). Applying Lemma 3.1, we obtain that
Eh(G
n
) = E [E [h(G
n
)U
n
]] = E
¯
h
n
(U
n
)
with
¯
h
n
(u) := Eh(G
k,Ψ
u
) for k = k(n). Thus we have to show that
E
¯
h
n
(U
n
) →
¯
h
∞
(1) := Eh(G
Ψ
). (37)
As h is bounded and U
n
a.s.
→ 1, the Continuous Mapping Principle (Billings
ley, 1968, Theorem 5.5) would yield (37) if we can prove that
¯
h
n
(u
n
) →
20
G. Mainik Estimating asymptotic dependence functionals
¯
h
∞
(1) for any sequence u
n
↑ 1. Thus, in order to verify (36), it suﬃces to
show that
G
k,Ψ
k
w
→ G
Ψ
in C(F
H,α
) (38)
for Ψ
k
:= Ψ
u
k
and u
k
↑ 1.
Recall that Ψ
u
w
→ Ψ as u ↑ 1 according to (13). Hence we have Ψ
k
w
→ Ψ.
Furthermore, Lemma 6.4 states that the function class F
H,α
is universally
Donsker and preGaussian, and F
H,α
is uniformly bounded according to
Remark 6.1(a). Thus, according to Lemma 2.8.7 from van der Vaart and
Wellner (1996), condition (38) is satisﬁed if
sup
f
1
,f
2
∈F
H,α
σ
Ψ
k
(f
1
− f
2
) − σ
Ψ
(f
1
− f
2
) → 0, (39)
where σ
Ψ
is the variance seminorm introduced in (20). Denote G := F
H,α
−
F
H,α
. As g 7→ σ
Ψ
(g) is continuous on G for any Ψ, condition (39) can also be
written as σ
Ψ
k
→ σ
Ψ
in C(G). Since G is compact (as a continuous image of
the compact F
H,α
×F
H,α
), σ
Ψ
k
→ σ
Ψ
in C(G) is equivalent to σ
Ψ
k
g
k
→ σ
Ψ
g
for g
k
→ g. This, however, easily follows from Ψ
k
g
2
k
→ Ψg
2
and Ψ
k
g
k
→ Ψg
due to Ψ
k
w
→ Ψ.
6.2 Proofs of the main results
This subsection contains the proofs of Theorems 3.2, 3.7, and 3.8
Proof of Theorem 3.2. Since bα
a.s.
→ α, we only need to consider ω ∈ Ω for
which bα is consistent. In this case we have that
sup
ξ∈H
f
ξ,bα
− f
ξ,α
∞
→ 0
for n → ∞, because the mapping (ξ, α, s) 7→ f
ξ,α
(s) is continuous and H
and S
d
1
are compact. Furthermore, each
b
Ψ is Lipschitz(1) as a mapping from
C(S
d
1
) to R (cf. Remark 6.1(b)). This yields
bγ
ξ
− γ
ξ
 ≤
f
ξ,bα
− f
ξ,α
∞
+
b
Ψf
ξ,α
− Ψf
ξ,α
.
Hence it suﬃces to prove that
b
Ψf
ξ,α
→ Ψf
ξ,α
uniformly in ξ ∈ H. However,
as the function class F
H,α
:= {f
ξ,α
: ξ ∈ H} is compact in (C(S
d
1
), k·k
∞
) and
all
b
Ψ are Lipschitz(1), uniform convergence of
b
Ψ on F
H,α
is equivalent to
the pointwise one.
It is obvious that
b
Ψf
ξ,α
− Ψf
ξ,α
≤
b
Ψf
ξ,α
− Ψ
U
n
f
ξ,α
+ Ψ
U
n
f
ξ,α
− Ψf
ξ,α
.
21
G. Mainik Estimating asymptotic dependence functionals
Thus, due to U
n
a.s.
→ 1 and Ψ
u
n
w
→ Ψ, it suﬃces to show that
b
Ψf
ξ,α
− Ψ
U
n
f
ξ,α
a.s.
→ 0. (40)
According to Lemma 3.1,
L
b
Ψf
ξ,α
− Ψ
U
n
f
ξ,α
U
n
= u
= L
k
X
j=1
1
k
(f
ξ,α
(Y
j
) − Ψ
u
f
ξ,α
)
with i.i.d. random vectors Y
1
, . . . , Y
k
∼ Ψ
u
. Denote Z
j
:= (f
ξ,α
(Y
j
) − Ψ
u
f
ξ,α
).
As f
ξ,α
is bounded, we have Z
j
 ≤ M for some M > 0. Thus Hoeﬀding’s
inequality (cf. Hoeﬀding, 1963, Theorem 2) yields
P
k
X
j=1
1
k
Z
j
> ε
≤ exp
−kε
2
/
2M
2
universally for all Ψ
u
, and hence
∞
X
n=1
P
n
b
Ψf
ξ,α
− Ψ
U
n
f
ξ,α
> ε
o
≤
∞
X
n=1
exp
−k(n)ε
2
/
2M
2
. (41)
By assumption we have that k(n) ≥ δn
q
for some q ∈ (0, 1) and δ > 0. This
gives a ﬁnite sum in (41), and the Borel–Cantelli Lemma yields (40).
Proof of Theorem 3.7. Part (a). Consider the decomposition
√
k (bγ
ξ
− γ
ξ
) =
√
k
P
n
f
ξ,bα
− Ψf
ξ,α
=
√
k
P
n
f
ξ,bα
− f
ξ,α
+ (P
n
− Ψ
U
n
) f
ξ,α
+ (Ψ
U
n
− Ψ) f
ξ,α
= P
n
h
√
k
f
ξ,bα
− f
ξ,α
i
+ G
n
f
ξ,α
+
√
k (Ψ
U
n
− Ψ) f
ξ,α
. (42)
First it should be noted that Condition 3.5(c) postulates
√
k (Ψ
U
n
− Ψ) f
ξ,α
P
→ b(ξ) in l
∞
(H)
and Lemma 6.5 implies
G
n
f
ξ,α
w
→ G
Ψ
f
ξ,α
in C(H).
Thus it suﬃces to consider the ﬁrst term in (42). Recall that the function
class ∂
2
α
F
H,α
is uniformly bounded according to Remark 6.1(a). Hence, for
any sequence y
n
→ y in R,
√
k
f
ξ,α+y
n
√
k
(s) − f
ξ,α
(s)
− ∂
α
f
ξ,α
(s)y
n
→ 0
22
G. Mainik Estimating asymptotic dependence functionals
uniformly in ξ and s, and thus in the space C(H × S
d
1
). This implies that
bg
ξ,k
:=
√
k
f
ξ,bα
− f
ξ,α
−
√
k∂
α
f
ξ,α
(bα − α)
P
→ 0
in C(S
d
1
) uniformly in ξ ∈ H, and P
n
bg ≤ P
n
kgk
∞
yields
P
n
h
√
k
f
ξ,bα
− f
ξ,α
i
−
√
k (bα − α) P
n
∂
α
f
ξ,α
P
→ 0
in C(H). Thus it suﬃces to show that
√
k (bα − α) P
n
∂
α
f
ξ,α
w
→ Y Ψ∂
α
f
ξ,α
(43)
in C(H) with a Gaussian random variable Y ∼ N(µ
α
, σ
2
α
). Since P
n
∂
α
f
ξ,α
P
→
Ψ∂
α
f
ξ,α
uniformly in ξ ∈ H (cf. Lipschitz(1) property of P
n
and compactness
of ∂
α
F
H,α
stated in Remark 6.1), (43) follows from the asymptotic normality
of bα. Finally, the asymptotic independence of the random variable Y
n
:=
√
k(bα − α) and the empirical process G
n
stated in Condition 3.5(b) yields
the result (21).
Part (b). This result is merely the ﬁnitedimensional version of Part (a).
It is easy to see that replacing the assumption (17) by (22) aﬀects only the
last term in (42) and results in an exchange of the uniform convergence to
b(ξ) by a pointwise version. Hence the pointwise asymptotic normality (23)
of bγ
ξ
follows immediately along the lines of the proof of Part (a).
Proof of Theorem 3.8. Part (a). Condition 3.5(b) and Theorem 3.7(a) imply
the joint convergence
√
k (bγ
ξ
− γ
ξ
, bα − α)
w
→ (Z, Y ),
where Y is the Gaussian limit in (15), independent of Z, and Z is the random
mapping in l
∞
(H) that appears on the right hand side of (21):
Z = Z(ξ) := b(ξ) + G
n
f
ξ,α
+ Ψ[∂
α
f
ξ,α
]. (44)
Moreover, the continuity of γ
ξ
, the compactness of H, and the assumption
γ
ξ
6= 0 imply that γ
ξ
∈ (ε, 1/ε) for all ξ ∈ H with some ε > 0. The
constant ε can always be chosen such that α ∈ (ε, 1/ε). Since the mapping
φ : (γ, α) 7→ γ
1/α
is C
2
on (0, ∞)
2
, the ﬁrstorder Taylor approximation of
φ(bγ
ξ
, bα) − φ(γ
ξ
, α) is uniform for (bγ
ξ
, bα) ∈ [ε, 1/ε]
2
, and (44) implies that
√
k (φ(bγ
ξ
, bα) − φ(γ
ξ
, α))
w
→ ∂
α
φ(bγ
ξ
, α)Y + ∂
t
φ(bγ
ξ
, α)Z.
The result follows from ∂
α
φ(t, α) = −α
−2
t
1/α
log t and ∂
t
φ(t, α) = α
−1
t
1/α−1
.
Part (b) is analogous, with obvious calculations.
23
G. Mainik Estimating asymptotic dependence functionals
6.3 Discussion of the regularity assumptions
Proof of Lemma 4.1. Due to (26) it suﬃces to show that
Ψ
U
n
− Ψ
1−k/n
f
ξ,α
= o
P
1/
√
k
uniformly in ξ ∈ H. Recall the notation Ψ
0
u
:= L(SF
R
(R) = u). Due to
F
R
(R) ∼ unif(0, 1), we have that
Ψ
u
f =
1
1 − u
Z
(u,1)
Ψ
0
v
f dv
for u ∈ [0, 1) and f ∈ F
H,α
. Thus continuity of the mapping u 7→ Ψ
0
u
f
ξ,α
entails diﬀerentiability of u 7→ Ψ
u
f
ξ,α
:
∂
∂u
Ψ
u
f
ξ,α
=
1
(1 − u)
2
Z
(u,1)
Ψ
0
v
f
ξ,α
dv −
1
1 − u
Ψ
0
u
f
ξ,α
=
1
1 − u
Ψ
u
− Ψ
0
u
f
ξ,α
.
Hence the MeanValue Theorem yields
Ψ
U
n
− Ψ
1−k/n
f
ξ,α
=
1
1 − u
∗
Ψ
u
∗
− Ψ
0
u
∗
f
ξ,α
(U
n
− (1 − k/n)) (45)
for some u
∗
between (1 −k/n) and U
n
. It is well known (cf. Smirnov, 1949)
that the random variable U
n
= F
R
(R
n:k+1
) satisﬁes
U
n
− (1 − k/n)
√
k/n
w
→ N(0, 1).
This implies U
n
− (1 − k/n) = O
P
√
k/n
, and therefore 1 − u
∗
= k/n +
O
P
√
k/n
. Consequently, (45) yields
Ψ
U
n
− Ψ
1−k/n
f
ξ,α
=
O
P
√
k/n
k/n + O
P
√
k/n
Ψ
u
∗
− Ψ
0
u
∗
f
ξ,α
= O
P
1/
√
k
Ψ
u
∗
− Ψ
0
u
∗
f
ξ,α
.
Hence it suﬃces to show that
sup
ξ∈H
Ψ
u
∗
− Ψ
0
u
∗
f
ξ,α
= o
P
(1). (46)
It is easy to see that
sup
ξ∈H
Ψ
u
∗
− Ψ
0
u
∗
f
ξ,α
= sup
ξ∈H
1
1 − u
∗
Z
(u
∗
,1)
Ψ
0
v