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FROM CONVEX COMBINATIONS OF REGULARLY
VARYING RANDOM VECTORS
Marco Oesting, Olivier Wintenberger
To cite this version:
Marco Oesting, Olivier Wintenberger. ESTIMATION OF THE SPECTRAL MEASURE FROM
CONVEX COMBINATIONS OF REGULARLY VARYING RANDOM VECTORS. 2020.
ESTIMATION OF THE SPECTRAL MEASURE FROM CONVEX COMBINATIONS OF REGULARLY VARYING RANDOM VECTORS
BYMARCOOESTING1,ANDOLIVIERWINTENBERGER2
1Stuttgart Center for Simulation Science and Institute of Stochatics and Applications, University of Stuttgart, 70569 Stuttgart,
Germany marco.oesting@mathematik.uni-stuttgart.de
2LPSM, Sorbonne Université, 4 place Jussieu, 75005 Paris, France, olivier.wintenberger@upmc.fr The extremal dependence structure of a regularly varying random vec-tor X is fully described by its limiting spectral measure. In this paper, we investigate how to recover characteristics of the measure, such as extremal coefficients, from the extremal behaviour of convex combinations of compo-nents of X. Our considerations result in a class of new estimators of moments of the corresponding combinations for the spectral vector. We show asymp-totic normality by means of a functional limit theorem and, focusing on the estimation of extremal coefficients, we verify that the minimal asymptotic variance can be achieved by a plug-in estimator.
1. Introduction. One of the main challenges in multivariate extreme value analysis is the inference of the dependence among the extremes of a random vector X = (X1, . . . , Xd)>∈
(0, ∞)dfrom a sample of independent copies X1, . . . , Xn. In risk management applications,
particular interest lies in the case that X possesses asymptotically dependent components, i.e. extremes tend to appear jointly in various components, with heavy tails. Many of these cases are covered by the framework of multivariate regular variation.
It is common practice in extreme value analysis to consider the extremal dependence structure of X separately from the marginal properties of the components X1, . . . , Xd. Thus, focus is
often put on the joint tail behaviour of some standardized vector X∗ obtained from X via marginal transformations. More precisely, the margins are standardized such that the tails are regularly varying with index 1 and asymptotically identical in the sense that
lim
x→∞
P(Xi∗> x)
P(Xj∗> x)= 1 for all i, j ∈ {1, . . . , d}.
By such a transformation, the property of multivariate regular variation is inherited from X to X∗. Thus, extremal dependence of X∗ can be fully described via its so-called spectral measure, that is, the limit distribution of the angular component X∗/kX∗k∞conditional on
kX∗k
∞> u as u → ∞.
In the past decades, the estimation of the spectral measure and related concepts has attracted a lot of attention, starting from the bivariate setting (d = 2), where estimation of the spectral measure is equivalent to inference for the so-called Pickands’ dependence function. Here, seminal estimation procedures come from Pickands [1975] with refinements such as Capéraà et al. [1997] using a copula approach assuming known margins. These procedures have been extended by Genest and Segers [2009] to the setting with unknown margins of X using the rank transform. All the above mentioned works were assuming that the observations are max-stable, i.e. that the asymptotic extreme regime is achieved, a typical assumption for ob-servations that correspond to maxima over certain temporal blocks such as annual maxima.
MSC 2010 subject classifications:Primary 62G32; secondary 62G30
Keywords and phrases:asymptotic statistics, extremal coefficient, moment estimator, multivariate extremes
The pioneer works of Einmahl et al. [1993] and Einmahl et al. [1997] deal with the esti-mation of the cumulative distribution function of the spectral measure with observations in the domain of attraction of max-stable distribution, including the case of regularly varying distributions. The two-step procedure in Einmahl et al. [1997] for dealing with unknown margins is replaced by a more efficient rank transform in Einmahl et al. [2001]. An alter-native approach due to Drees and Huang [1998] focuses on the estimation of the stable tail dependence function
L(x) = Eh max
i=1,...,dxiΘi
i
, x = (x1, . . . , xd) ∈ (0, ∞)d,
using the rank transform for d = 2. This work was extended to the multivariate case d > 2 in Einmahl et al. [2012]. We refer to Chapter 7 of the monograph De Haan and Ferreira [2006] for a general overview of non-parametric approaches.
When d is large, any inference of the complete spectral distribution suffers from the curse of dimensionality. Thus, it is more reasonable in practice to focus on the joint tail behaviour of the components of lower-dimensional subvectors XI∗= (Xi∗)i∈I for sufficiently many
“di-rections” I that are non-empty subsets of {1, . . . , d}. In this framework, one often makes use of summary statistics such as the so-called extremal coefficients [cf. Smith, 1990, Schlather and Tawn, 2003] given by
(1) τI= lim
u→∞
P(maxi∈IXi∗> u)
P(Xj∗> u)
, j ∈ I ,
a number between 1 and |I|. Such parameter has attracted a lot of attention as it quantifies the effective number of asymptotically independent components among I. Some properties, such as self-consistency for sets of extremal coefficients, and various procedures for esti-mating extremal coefficients were introduced by Schlather and Tawn [2003]. For max-stable random fields, Cooley et al. [2006] showed that inference can also be based on the mado-gram, a graphical tool originally developed in geostatistics.
In this paper, we also consider the extremal dependence structure of subvectors XI∗ de-scribed in terms of the corresponding spectral vector ΘI, that is, the limit in distribution of XI∗/kXI∗k∞ conditional on `I(X∗) := kXI∗k∞> u as u → ∞. While existing works
typically focus on a direct estimation of the distribution of ΘI, i.e. the spectral measure, we focus on moment estimators of ΘI. Our approach further differs from previous works since we impose the assumption that the original observations X are standard regularly vary-ing with a unique tail index α > 0 and tail equivalent margins. This restriction allows us to consider a two-step procedure in the spirit of Einmahl et al. [1997] with a simpler first standardization step relying on the componentwise normalization via upper order statistics and the estimation of a unique coefficient α > 0 via a Hill type estimator. Our second step is then based on the estimation of moments of convex combinations of the spectral vector, i.e. estimators of E[(v>ΘI)p] with p ≥ 1 and v on the simplex of the direction I. We prove that the collection of such moments E[(v>ΘI)p], p ≥ 1, is characterizing the spectral mea-sure of XI∗. Perceiving the resulting estimators as functions of the weight vector v on the simplex, we obtain functional central limit theorems with precise asymptotic (co-)variances. Here, we consider both the case that the margins of X are known, i.e. we can directly work with exactly standardized observations X∗, and the case that the margins are unknown and a preliminary empirical standardization step has to be included.
In accordance to the above-mentioned paradigm that, in large dimensions d, emphasis is often put on simpler summary statistics, we suggest to mainly focus on small p for which our approach provides simple functions that contain important information on the depen-dence among extremes. For instance, by tail equivalence of the marginals of X∗we have that
E[ΘIi] = E[ΘIj] = 1/τI for any i, j ∈ I and, consequently, for any positive weights v such
thatP
i∈Iv1= 1 we have that E[vTΘI] = 1/τI. Thus, the moment of order p = 1 of convex
combinations of ΘI does not depend on the weight vector v. This observation motivates an additional third step of our procedure. More precisely, similarly to Mainik and Rüschendorf [2010] who estimate the optimal portfolio of jointly regularly varying assets, we use the func-tional central limit theorem to estimate the weight vector v∗ that minimizes the asymptotic variance and show that this minimal asymptotic variance can be achieved by a plug-in esti-mator. We note that the determination of v∗is interesting on its own and provides important information on the dependence among extremes. The determination of a prominent direction in the data is related to recent developments on extreme PCA in Cooley and Thibaud [2019] and Drees and Sabourin [2019]. Moreover we discuss how the weights v∗ are impacted by the fact that the margins are known or not, even in our setting of standard regular variation where the extremes in different directions are comparable.
The paper is organized as follows. In Section 2, we gather several results on convex combina-tions of the spectral components of a regularly varying vector. The asymptotic normality of empirical moment estimators is given in Section 3 both for the case of known and the case of unknown margins. These results specified to first order moments are then applied in Section 4 in order to obtain estimators for extremal coefficients with minimal asymptotic variance. Some comparison to benchmark estimators based on the definition of τIand some illustration
are provided. We conclude with a short discussion on our results in Section 5.
2. Extremes of convex combinations of regularly varying random vectors. Let X = (X1, . . . , Xd)>be a regularly varying E = [0, ∞)d-valued random vector, that is, there exists
a regularly varying function a : (0, ∞) → (0, ∞) such that
a(u)P(u−1X ∈ ·)−→ µ(·) ,v u → ∞ ,
for a measure µ on E \ {0} which is homogeneous of order −α < 0, the so-called exponent measure. In the following, we will assume that the exponent measure is non-degenerate, i.e. (2) ri:= µ({x ∈ E : xi> 1}) > 0 for all i = 1, . . . , d.
In particular, a(u)P(Xi> u) → ri as u → ∞ for any 1 ≤ i ≤ d and thus
P(Xi> u)
P(Xj> u)
−→ ri rj
∈ (0, ∞) , u → ∞ ,
for all i, j ∈ {1, . . . , d}, i.e. X1, . . . , Xd are tail equivalent. Assuming that the index α > 0
and the scaling factors r1, . . . , rd> 0 are known, we will directly focus on the transformed
vector X∗= (r−11 X1α, . . . , r−1d Xdα) which is necessarily regularly varying with index 1. With a∗(u) = a(u1/α), we obtain that
(3) a∗(u)P(u−1X∗∈ ·)−→ µv ∗(·) , u → ∞ ,
for some normalized exponent measure µ∗ which is homogeneous of order −1 and satisfies (4) µ∗({x ∈ E : xi> 1}) = lim
u→∞a ∗
(u)P(Xi∗> u) = 1,
i.e. a∗(u) ∼ P(Xi∗ > u)−1 for all i = 1, . . . , d. This verifies that the tail behaviour of all
the components of X∗ is asymptotically identical, i.e. its marginal distribution functions F1∗, . . . , Fd∗satisfy
1 − Fi∗(u)
for all i, j ∈ {1, . . . , d}, which also implies
(5) (F
∗ i)−1(p)
(Fj∗)−1(p)→ 1, p → 1.
Equation (3) can be equivalently expressed in terms of a spectral component for an arbitrary norm k · k on Rd. More precisely, we obtain the weak convergence
(6) L(u−1X∗| kX∗k > u)−→ L(Y Θ) ,w u → ∞ ,
where Y is a unit Pareto random variable and, independently from Y , Θ is a [0, ∞)d-valued random vector satisfying kΘk = 1 almost surely, i.e.
P(Θ ∈ Sd−1+ ) = 1, where Sd−1+ = {x ∈ [0, ∞)d: kxk = 1}.
In the following, we will focus on the maximum norm k · k = k · k∞ which is not a serious
restriction as all norms on Rdare equivalent.
There is a one-to-one correspondence between the normalized exponent measure µ∗ and the distribution of Θ, the so-called (normalized) spectral measure (w.r.t. the norm k · k) via the relation
(7) µ∗({x ∈ E : kxk > r, x/kxk ∈ B}) = τ r−αP(Θ ∈ B), r > 0, B ⊂ Sd−1+ for some constant 1 ≤ τ ≤ d given by
(8) τ = µ∗({x ∈ E : kxk > 1}) = lim
u→∞a ∗
(u)P(kX∗k > u) .
Thus, the extremal dependence structure of the components of the normalized random vector X∗, and, thus, also of the original vector X, is fully described by the distribution of the spectral vector Θ which we will estimate in the following.
More precisely, we now describe the tail behaviour of convex combinations of X1∗, . . . , Xd∗in terms of the corresponding spectral measure. To this end, we will make use of the following well-known lemma which is a direct consequence of the weak convergence in (6).
LEMMA 1. LetX∗,Y and Θ be as above. Furthermore, let g : (0, ∞) × Sd−1+ → R be a continuous bounded function. Then, we have
lim
u→∞Eg(u
−1kX∗k, X∗
/kX∗k) | kX∗k > u = E[g(Y, Θ)].
In particular, the lemma implies that
(9) lim
u→∞E[˜g(X
∗/kX∗k) | kX∗
k > u] = E[˜g(Θ)] for any continuous function ˜g : Sd−1+ → R.
Furthermore, the lemma extends to indicator functions of sets that are continuous with respect to the measure of (Y, Θ), i.e. conditional probabilities of events that can be expressed in terms of u−1kX∗k and X∗/kX∗k. For instance, for convex combinations of X1∗, . . . , Xd∗ with coefficient vector v ∈ B1(0) = {x ∈ Rd: kxk1≤ 1}, v 6= 0, we obtain
lim u→∞P(v > X∗> u | kX∗k > u) = PY v>Θ > 1 (10) = E h (v>Θ)+∧ 1 i = E h (v>Θ)+ i
using that v>Θ ≤ kvk1· kΘk = kvk1≤ 1 a.s. It can be shown that the law of Θ is uniquely
determined by the set of all the limits in (10) with v ∈ ∂B1(0) = {x ∈ Rd: kxk1 = 1},
see Basrak et al. [2002], Klüppelberg and Pergamenchtchikov [2007], Boman and Lindskog [2009]. This fact motivates the use of extremes of convex combinations of observations for inference of the spectral measure. In this paper, we modify the above approach by assessing the limiting quantity on the RHS of Equation (10) in an alternative way based on the following result which is a trivial consequence of Lemma 1.
LEMMA 2. Let X∗ be a regularly varying random vector with standard Pareto margins. Then, for anyv ∈ Rdandp ∈ N, we have
(11) E v> X ∗ kX∗k p + kX∗k > u −→ E[(v>Θ)p+] asu → ∞.
PROOF. The result directly follows by applying Equation (9) to the continuous functions ˜
gv,p(θ) = (v>θ)p+with v ∈ Rdand p ∈ N.
Note that, for v 6≥ 0, the term v>Θ is negative with positive probability, which hampers the calculation of closed-form expressions for the RHS of (11). In order to avoid these difficulties, we restrict our attention to non-negative vectors v ∈ ∂B1+(0) = ∂B1(0) ∩ [0, ∞)d, i.e. we
consider only convex combinations of components of X∗ with positive coefficients, while allowing for arbitrary powers of these combinations. This ensures the identifiability of the spectral measure from the resulting limits as the following proposition shows.
PROPOSITION3. LetX∗be a non-negative regularly varying random vector with index 1.
Then, the spectral measure ofΘ is uniquely determined by the moments
(12) nE[(v>Θ)p+] : v ∈ ∂B1+(0), p ∈ N
o
=nE[(v>Θ)p] : v ∈ ∂B1+(0), p ∈ N
o .
PROOF. By a series expansion, for all s ∈ [0, ∞)d, the limits in (12) uniquely determine
E[exp(−s>Θ)] = ∞ X p=0 (−1)p·ksk p 1 p! · E s> ksk1Θ p , s ∈ [0, ∞),
i.e. the Laplace transform of the spectral measure.
Let us list some properties of the function v 7→ mp(v) = E[(v>Θ)p] for p = 1, 2, . . . and
v ∈ ∂B1+(0). We identify m1(v) =Pdi=1viE[Θi], i.e. the function is linear for p = 1. As,
for all i = 1, . . . , d, E[Θi] = Z 1 0 P(Θi> x) dx = Z ∞ 1 y−2P(Θi> 1/y) dy = P(Y Θi> 1) = lim u→∞P(X ∗ i > u | kX ∗k > u) = 1 a∗(u)P(kX∗k > u)= 1 τ, (13)
where τ is defined as in Equation (8), we even obtain m1(v) =τ1Pdi=1vi, which implies that
m1is constant on ∂B+1(0). For p ≥ 2, the function mp is strongly convex. Thus, one easily
deduces from Jensen’s inequality that
mp(v) ≤ max 1≤i≤dE[Θ
p i].
For many popular limit models, closed-form expressions for E[(v>Θ)p] are difficult to obtain as the spectral measure w.r.t. the maximum norm k · k = k · k∞is often difficult to handle in
high dimensional settting. Instead, spectral vectors are often normalized w.r.t. some less com-plex functional `I(x) =Wi∈Ixi over some smaller subset I ⊂ {1, . . . , d}. Such functionals
`I play an important role when asserting extremal coefficients, see Section 4. Then, we have
that ∂{x ∈ E : `I(x) > 1} ⊂ {x ∈ E : `I(x) = 1} is a µ∗-null set and, thus, the limit
(14) τI:= µ∗({x ∈ E : `I(x) > 1}) = lim u→∞a
∗
(u)P(`I(X∗) > u)
exists and, by Equation (3), equals µ∗({x ∈ E : `I(x) > 1}). Furthermore, for any I 6= ∅, it
satisfies the relation
(15) τI= E[`I(Θ)] · τ ∈ [1, |I|].
Since E[`I(Θ)] > 0 as soon as I 6= ∅, we can study the behaviour of the `I-spectral vector
ΘIwhose distribution can be defined from the original spectral measure via the relation
(16) P ΘI∈ A = 1
E[`I(Θ)]
Z
[0,∞)d
1{θ/`I(θ) ∈ A}`I(θ) P(Θ ∈ dθ),
where A ⊂ {w ∈ [0, ∞)d: `I(w) = 1}. Note that, from Equation (15) and Equation (16), we
directly obtain an analogous result to Equation (13), namely
(17) E[ΘIi] =
1 τI
, i = 1, . . . , d.
Analogues to Equation (10) and to Lemma 2 are given in the following proposition.
PROPOSITION4. LetX∗ be a normalized regularly varying random vector with index1.
Then, for anyv ∈ Rdandk ∈ N, we have
P v>X∗> u | `I(X∗) > u −→ E v>ΘI +∧ 1 =E(v >Θ) +∧ `I(Θ) E[`I(Θ)] (18) E v> X ∗ `I(X∗) p + `I(X∗) > u −→ E v>ΘI p + = E v>`Θ I(Θ) p +· `I(Θ) E [`I(Θ)] (19) asu → ∞.
PROOF. First, we note that by definition of `I we `I(x) ≤ kxk on [0, ∞)d\ {0}. Thus
`I(X∗) > u implies that kX∗k > u and we have
lim u→∞P v>X∗> u | `I(X∗) > u = lim u→∞ P v>X∗> u, `I(X∗) > u | kX∗k > u P (`I(X∗) > u | kX∗k > u) =P Y (v >Θ) > 1, Y ` I(Θ) > 1 P (Y `I(Θ) > 1) =E(v >Θ) +∧ `I(Θ) ∧ 1 E [`I(Θ) ∧ 1] = 1 E[`I(Θ)]· E h (v>Θ)+∧ `I(Θ) i = E v>ΘI + ∧ 1 ,
where we used the fact that `I(Θ) ≤ kΘk = 1 and we apply the change of measure (16).
g(y, θ) = (v>θ)p+1{`I(yθ) > 1} whose discontinuity set {`I(yθ) = 1} has no limiting mass,
i.e. P(`I(Y Θ) = 1) = P(Y `I(Θ) = 1) = 0, we further obtain
lim u→∞E v> X ∗ `I(X∗) p + `I(X∗) > u = lim u→∞ E v>`X∗ I(X∗) p +1{`I(X ∗) > u] kX∗k > u P (`I(X∗) > u | kX∗k > u) = E v>` Θ I(Θ) p +1{Y `I(Θ) > 1} P (Y `I(Θ) > 1) = E v>` Θ I(Θ) p + · (`I(Θ) ∧ 1) E [`I(Θ) ∧ 1] = 1 E [`I(Θ)]· E v> Θ `I(Θ) p + · `I(Θ) = E v>ΘIp + .
Similarly to Proposition 3, it can be shown that the distribution of ΘIis uniquely determined by the limits n E[(v>ΘI)p+]; v ∈ ∂B1+(0), p ∈ N o =nE[(v>ΘI)p]; v ∈ ∂B1+(0), p ∈ N o .
While the `I-spectral measure, i.e. the distribution of ΘI, can be written in terms of the
original spectral measure as in (16), the original spectral measure can be uniquely recovered from the distribution of ΘI only if P(`I(Θ) = 0) = 0. Then, the inverse transformation is
given by (20) P (Θ ∈ A) = 1 E[kΘIk] Z [0,∞)d 1{θ/kθk ∈ A}kθk P(ΘI∈ dθ), A ⊂ Sd−1+ .
Note that the discussion above can be easily extended to any positively homogeneous contin-uous functional ` : [0, ∞)d→ [0, ∞) with corresponding constant τ` which is positive.
3. Asymptotic normality.
3.1. Asymptotic normality for deterministic thresholds. Assume that we observe indepen-dent copies Xl∗= (Xl1∗, . . . Xld∗)>, l = 1, . . . , n, of the normalized regularly varying random vector X∗. Consider a non-empty subset I ⊂ {1, . . . , d}. One of the usual approaches to es-timate the distribution of the `I-spectral vector ΘI is based on the conditional exceedance
probabilities of convex combinations given in relation (18), i.e.
lim u→∞P(v >X∗> u | ` I(X∗) > u) = E h (v>ΘI)+∧ 1 i
for v ∈ ∂B1(0) = {x ∈ Rd: kxk1= 1}. The associated empirical estimators are of the form
b Pn,u,I(conv)(v) b Pn,u,I =n −1Pn l=11{v>Xl∗> u, `I(X ∗ l) > u} n−1Pn l=11{`I(Xl∗) > u} , v ∈ ∂B1(0) . (21)
In view of Proposition 3, we promote in this paper the use of the alternative characterization of the distribution of the `I-spectral vector ΘI based on the limits of moments of convex
combinations provided by Equation (19), i.e.
lim u→∞E v> X ∗ `I(X∗) p `I(X∗) > u = Eh(v>ΘI)pi,
for any v ∈ ∂B1+(0) and p ∈ N0. The associated empirical estimators are of the form (22) Mcn,u,I(v, p) b Pn,u,I = n−1Pn l=1 v> Xl∗ `I(Xl∗) p 1{`I(Xl∗) > u} n−1Pn l=11{`I(Xl∗) > u} , v ∈ ∂B1+(0).
In the following, we will consider the new estimators cMn,u,`(v, p)/ bPn,u,` for any p ∈ N0.
As the functional `I puts emphasis on the components (Xi∗)i∈I of X∗ only, we will focus
on the same components in the convex combination, i.e. we will consider weight vectors vI where vI is the vector with components vi for i ∈ I and 0 otherwise. Thus, we obtain
vI>X∗=P
i∈IviX ∗ i.
The next theorem will provide a functional central limit theorem for the estimators c
Mn,u,I(vI, p) noting that bPn,u,I = cMn,u,I(vI, 0) for all v ∈ ∂B1+(0). Then we can
ap-ply the functional delta method to obtain the asymptotic result for the ratio estimator c
Mn,u,I(vI, p)/ bPn,u,I.
While, in this subsection, we assume observations from the normalized random vector X∗, in Subsection 3.2 the more general case of observations from the regularly varying vector X with unknown index α > 0 and unknown marginal scales is considered. To be disposed for this extension, in the next theorem, we do not only prove a functional central limit theorem for cMn,u,I(vI, p) with v ∈ ∂B1+(0) and p ∈ N0, but already for a more general version.
Indeed, it will turn out to be useful to allow for multiplication of single components of X∗ by a vector s ∈ [0, ∞)d of factors and for raising them up to a power of 1/β > 0. In the following, let (s ◦ X∗)1/β= (s1/βi Xi∗1/β)di=1. With this notation, we can write
`I(X∗) = max i∈I |X
∗
i| = k1I◦ X∗k,
where 1 = (1, . . . , 1) ∈ Rd, i.e., according to the notation introduced above, 1I is the vector
with the ith component being equal to 1 if i ∈ I and being equal to 0 otherwise. Furthermore, vI>X∗= v>(1I◦ X∗). Thus, we can rewrite
(23) Mcn,u,I(vI, p) = 1 n n X l=1 v> 1I◦ X ∗ l k1I◦ Xl∗k p 1{k1I◦ Xl∗k > u},
with the right-hand-side of the equation depends on the components vi, i ∈ I, of v only. In
our extended version of the functional central limit theorem, we will not restrict our attention to the vector 1I◦ X∗ being extreme, but a perturbed vector (s ◦ X∗)1/β with s being close
to 1Iand β being close to 1. More precisely, we consider s ∈ Aδwhere
Aδ,I =
n
s ∈ (0, ∞)d: si= 0 for all i /∈ I, (1 + δ)−1≤ si≤ 1 + δ for all i ∈ I
o
for some δ ≥ 0, i.e. we allow for a perturbation of a factor between (1 + δ)−1 and 1 + δ in each component of s with index i ∈ I and β. We also denote the disjoint union
Aδ=
[
∅6=I⊂{1,...,d}
Aδ,I.
Replacing 1I◦ X∗in Equation (23) by (s ◦ X∗)1/β, we obtain the generalized estimator
c Mn,u(v, s, β, p) = 1 n n X l=1 v> (s ◦ X ∗ l)1/β k(s ◦ Xl∗)1/βk !p 1{ks ◦ Xl∗k > u},
for v ∈ ∂B1+(0), (s, β) ∈ A0δ and p ∈ N0, where A0δ = Aδ × [(1 + δ)−1, (1 + δ)−1].
By construction, cMn,u,I(vI, p) = cMn,u(vI, 1I, 1, p) for every I ⊂ {1, . . . , d}. In
be read off from the components of s that are strictly positive. Analogously, we define b
Pn,u(s) = cMn,u(v, s, β, 0) which depends neither on v ∈ ∂B1+(0) nor on β > 0, and
sat-isfies bPn,u,I= bPn,u(1I).
Applying Proposition 4, we obtain
a∗(u)E h c Mn,u(v, s, β, p) i = a∗(u)P(ks ◦ X∗k > u) · E " v> (s ◦ X ∗ l)1/β k(s ◦ X∗ l)1/βk !p ks ◦ X∗k > u # −→ u→∞τ`· E h v> ΘI1/βpi = τ · E " v> s ◦ Θ ks ◦ Θk 1/β!p ks ◦ Θk # , (24)
where we used Equation (15) and Equation (16). Beyond the limiting behaviour of the ex-pectation, the following theorem establishes asymptotic normality of cMn,u(v, 1I, 1, p). A
detailed proof is given in Appendix A.
THEOREM5. LetXl∗,l ∈ N, be independent copies of a regularly varying [0, ∞)d-valued
random vector X∗ with index α = 1 satisfying Equation (4) and denote the correspond-ing spectral vector by Θ. Furthermore, let un→ ∞ such that n/a∗(un) → ∞. Then, for
any finite set K0 ⊂ N0 and any δ ≥ 0, the sequence of processes ({Gn(v, s, β, p); v ∈
∂B1+(0), (s, β) ∈ A0δ, p ∈ K0})n∈Nwith Gn(v, s, β, p) = r n a∗(u n) h a∗(un) cMn,un(v, s, β, p) − a ∗ (un)E h c Mn,un(v, s, β, p) ii (25)
converges weakly in`∞(∂B+1(0) × A0δ× K0) to a tight centered Gaussian process G with
covariance Cov(G(v, s, β1, p1), G(w, t, β2, p2)) (26) = τ E " v> s ◦ Θ ks ◦ Θk 1/β1!p1 w> t ◦ Θ kt ◦ Θk 1/β2!p2 (ks ◦ Θk ∧ kt ◦ Θk) # .
A direct consequence of the weak convergence of Gnis that a∗(un) cMn,un(v, s, β, p)
con-verges in probability to the limit expression in Equation (24). Consequently, the ratio estima-tor cMn,un(v, s, β, p)/ bPn,un(s) is consistent:
(27) Mcn,un(v, s, β, p) b Pn,un(s) =Mcn,un(v, s, β, p) c Mn,un(v, s, β, 0) →pc(v, s, β, p)
for all v ∈ ∂B1+(0), (s, β) ∈ A0δand p ∈ N0 where
c(v, s, β, p) := 1 E[ks ◦ Θk]· E " v> s ◦ Θ ks ◦ Θk 1/β!p · ks ◦ Θk # .
Applying Theorem 5 and using the functional Delta method, we can even establish the de-sired functional limit theorem for the ratio estimator cMn,un(v, s, β, p)/ bPn,un(s) provided
that we can neglect the preasymptotic bias that arises from the fact that Equation (24) yields an asymptotic relation only.
COROLLARY 6. Let the assumptions of Theorem 5 hold and assume that there exist some δ ≥ 0 and some finite subset K ⊂ N such that, for all p ∈ K ∪ {0},
(28) r n a∗(u n) a ∗(u
n)E[ cMn,un(v, s, β, p)] − τ E[ks ◦ X
∗k] c(v, s, β, p) → 0 uniformly inv ∈ ∂B1∗(0) and (s, β) ∈ A0δ.
Then, the sequence of processes ({ eGn(v, s, β, p); v ∈ ∂B1+(0), (s, β) ∈ A0δ, p ∈ K})n∈N
with e Gn(v, s, β, p) = r n a∗(u n) c Mn,un(v, s, β, p) b Pn,un(s) − c(v, s, β, p) ! ,
converges weakly in`∞(∂B1+(0) × A0δ× K) to a tight centered Gaussian process eG. The proof of Corollary 6 is deferred to Appendix B. The expression of the covariance of eG is provided there in Equation (51).
REMARK7. 1. For the original estimator with deterministic threshold as given in (22), it
is sufficient to restrict to the case that s = t = 1Iand β = 1 in Corollary 6. Then, we have
ks ◦ Θk∞= kt ◦ Θk∞= `I(Θ) and the covariance of eG given in (51) simplifies to
CovG(v, 1e I, 1, p1), eG(w, 1I, 1, p2) = 1 τ E`I(Θ) · 1 E`I(Θ)E v> ΘI `I(Θ) p1 · w> ΘI `I(Θ) p2 · `I(Θ) − 1 (E`I(Θ))2E v> ΘI `I(Θ) p1 · `I(Θ) E w> ΘI `I(Θ) p2 · `I(Θ) = 1 τI Cov (v>ΘII)p1, (w>ΘI I)p2 .
2. Using the same techniques as in the proof of Corollary 6, one can easily show the joint weak convergence of (Gn(v, s, β1, p1), eGn(w, t, β2, p2))n∈N for any (v, s, β1, p1),
(w, t, β2, p2) in ∂B1+(0) × A0δ× K to (G(v, s, β1, p1), ˜G(w, t, β2, p2)) which are jointly
Gaussian. with covariance
Cov(G(v, s, β1, p1), eG(w, t, β2, p2)) =E h v>ks◦Θks◦Θ 1/β1p1w>kt◦Θkt◦Θ 1/β2p2(ks ◦ Θk ∧ kt ◦ Θk)i E[kt ◦ Θk] −E h v>ks◦Θks◦Θ 1/β1p1(ks ◦ Θk ∧ kt ◦ Θk)iE h w>kt◦Θkt◦Θ 1/β2p2kt ◦ Θki (E[kt ◦ Θk])2 .
3. From the proofs of Theorem 5 and Corollary 6, it follows that, for fixed p ∈ N0, the
se-quences of processes Gn(·, ·, ·, p) and eGn(·, ·, ·, p) are asymptotically uniformly
equicon-tinuous on ∂B1+(0) × A0δw.r.t. the semimetric
ρ((v, s, β1), (w, t, β2)) = max{kv − wk, ks − tk, |β1− β2|}.
Consequently, almost all sample paths of the tight limits G and eG, respectively, are uni-formly ρ-continuous [cf. Addendum 1.5.8 in Van der Vaart and Wellner, 1996]. Since ρ induces the standard topology on ∂B+1(0) × A0δ as subset of R|I|× Rd+1, it follows that
3.2. Asymptotic normality for random thresholds via order statistics. Consider the general case of a random vector X that is regularly varying and satisfies Equation (2). In practice, the marginal distributions of X1, . . . , Xdare unknown. In particular, the index α of regular
varia-tion and the scaling factors r1, . . . , rdas defined in (2) that are needed for the transformation
to the normalized vector X∗ cannot be assumed to be given. Furthermore, the normalizing function a∗(·) ∼ P(Xi∗> ·) is unknown. To cope with these difficulties, we propose to work
with an estimatorα of α and order statistics of X. More precisely, let Xb k:n,i denote the kth upper order statistic of X1i, . . . , Xnifor i = 1, . . . , d and let Xk:n= (Xk:n,1, . . . , Xk:n,d). We
propose the following estimator
f Mn,k,I(v, p) e Pn,k,I = n−1Pn l=1 vI> Xl/Xk:n `I(Xl/Xk:n) αb p 1{`I(Xl/Xk:n) > 1} n−1Pn l=11{`I(Xl/Xk:n) > 1}
for some k < n, where the ratio of vectors Xl/Xk:n is to be interpreted componentwise.
Expanding each of the ratios Xl/Xk:n by r1−1, . . . , r −1
d , respectively, and taking the power
of α, it can be easily seen that
f Mn,k,I(v, p) e Pn,k,I = n−1Pn l=1 vI> Xl∗/X∗k:n `I(Xl∗/Xk:n∗ ) α/αb p 1{`I(Xl∗/Xk:n∗ ) > 1} n−1Pn l=11{`I(Xl∗/Xk:n∗ ) > 1} ,
i.e. we can rewrite
(29) Mfn,k,I(v, p) e Pn,k,I =Mcn,un(vI, un/X ∗ k:n◦ 1I, α/α, p)b b Pn,un(un/X ∗ k:n◦ 1I) .
For an appropriate sequence {kn}n∈N ⊂ N with kn→ ∞ such that kn/n → 0, we want to
show that ( p kn f Mn,kn,I(v, p) e Pn,kn,I − E[(vI>ΘII)p] ! ; v ∈ ∂B1+(0) )! n∈N
converges weakly to a Gaussian process. The functional central limit theorem will essentially be proven by plugging the random scaling factor s = un/Xk:n∗ ◦ 1I and the random power
β = α/α into the results of Subsection 3.1 according to Equation (29). Provided that bothb the random s and the random β are asymptotically normal, we can use the functional central limit theorem given in Corollary 6 to obtain the desired result.
To this end, we will first specify our estimator α of α which will be a Hill type estimator.b Similarly to Section 3.1, we will first start with an estimator with a fixed sequence of thresh-olds {un}n∈N and normalized observations X1∗, . . . , Xn∗. To mimic the behaviour of the real
non-normalized data X = (r ◦ X∗)1/αwhere r = (r
1, . . . , rd), we also allow for a scaling
factor s ∈ Aδ and take the norm of the scaled vector to the power of 1/α. Thus, we obtain
the estimator (30) Lbn,u(s) b Pn,u(s) = n−1Pn l=1log s ◦ X∗ l un 1/α 1{ks ◦ Xl∗k > un} b Pn,u(s) , s ∈ Aδ.
Under the assumptions of Theorem 5, it can be shown that
a∗(un)E[bLn,un(s)] ∼ τ E[ks ◦ Θk] ·
1 αE h log s ◦ X∗ un ks ◦ X ∗k > u n i → τ E[ks ◦ Θk] · 1 α
as n → ∞. If we assume that the bias is negligible, i.e. (31) sup s∈Aδ r n a∗(un)
a∗(un)E[bLn,un(s)] − τ E[ks ◦ Θk]
1 α = 0,
we can establish asymptotic normality for our final Hill type estimator αbk,n,I defined via order statistics (32) 1 b αn,k,I =n −1Pn l=1log (`I(Xl/Xk:n)) 1 {`I(Xl/Xk:n) > 1} e Pn,k,I
in the following theorem. A detailed proof is given in Appendix C.
THEOREM8. LetXl,l ∈ N, be independent copies of a regularly varying [0, ∞)d-valued
random vector X satisfying Equation (2). Denote the corresponding transformed vector by X∗ = (r−11 X1α, . . . , r−1d Xdα) and its marginal distribution Fi∗ for all 1 ≤ i ≤ d. Let {kn}n∈N ⊂ N be a sequence such that kn → ∞. If a sequence un∼ (Fi∗)−1(1 − kn/n)
(which automatically holds for all i ∈ {1, . . . , d} simultaneously by Equation (5)) satisfies the assumptions of Theorem 5 as well as Equation (31) and Equation (28) for p = 0 and someδ > 0. Then we have
p kn 1 b αn,kn,I − 1 α →dHeI, n → ∞, I ⊂ {1, . . . , d}, I 6= ∅,
where eHIis a centered Gaussian random variable with variance1/(τIα2).
REMARK9. We can show the joint convergence of the different estimates to the limit
pro-cesses G and eG and the variables eHI, I ⊂ {1, . . . , d}, as obtained in Theorem 5, Corollary
6 and Theorem 8, respectively. The covariance between the processes G and eG is given in the second part of Remark 7. Similar calculations as above reveal that, for all v ∈ ∂B1+(0), s ∈ Aδ, β ∈ [(1 + δ)−1, 1 + δ] and p ∈ N0, Cov( eHI, G(v, s, β, p)) (33) =E h − log1 ∧ k1ks◦Θk I◦Θk ·v> s◦Θ ks◦Θk 1/βp · (k1I◦ Θk ∧ ks ◦ Θk)i αE[k1I◦ Θk] .
For the covariance between eHI and eG(v, s, β, p), we are particularly interested in the case
that s = 1Ifor the same index set I. Here, using similar arguments as above, we obtain
(34) Cov( eHI, eG(v, 1I, β, p)) = 0
for all v ∈ ∂B1+(0), β ∈ [(1 + δ)−1, 1 + δ] and p ∈ N0.
Making use of the asymptotic results for the Hill-type estimator in Theorem 8 and for the estimators in the case of deterministic thresholds given in Section 3.1, we will establish a functional central limit theorem for fMn,k,I(vI, p)/ ePn,k,I. Compared to its analogue for
de-terministic thresholds, the estimator is blurred by the random marginal normalization based on the order statistics. To ensure that the estimator is unbiased, an additional assumption is required which can be interpreted as a second order condition on the marginal distributions. It corresponds to condition (28) for s ∈ Aδ,{i}and p = 0, i.e. it ensures the negligibility of the
presasymptotic bias in the convergence of Gn(·, s, ·, 0) to G(·, s, ·, 0) for s ∈ Aδ,{i}. Remind
the limit process G(·, s, ·, 0) are constant on ∂B+1(0) × [(1 + δ)−1, 1 + δ]. Thus, to ease notation, we just write G0(s) = G(·, s, ·, 0).
More precisely, we assume that
(35) sup
(1+δ)−1≤s≤1+δ
p
kn|a∗(un)P(sXi∗> un) − s| → 0 , n → ∞.
for some δ > 0 and un∼ (Fi∗)−1(kn/n). Imposing some additional conditions on the
func-tion c, we obtain the following main result.
THEOREM10. LetXl,l ∈ N, be independent copies of a regularly varying [0, ∞)d-valued
random vector X satisfying Equation (2) such that the corresponding transformed vec-tor X∗= (r1−1X1α, . . . , r−1d Xdα) and the sequence {kn}n∈N⊂ N satisfy the assumptions of
Corollary 6 and Theorem 8 for someδ > 0 and some K ⊂ N and Equation (35) holds for all i ∈ {1, . . . , d}. Assume that, for all p ∈ K, all the partial derivatives
ci(v, s, β, p) = ∂ ∂si c(v, s, β, p), i ∈ {1, . . . , d}, cβ(v, s, β, p) = ∂ ∂βc(v, s, β, p),
exist and are continuous on∂B1+(0) × A0δ. Then, for any non-emptyI ⊂ {1, . . . , d}, (√ k Mfn,k,I(vI, p) e Pn,k,I − c(vI, 1I, 1, p) ! ; v ∈ ∂B+1(0), p ∈ K ) −→nG(ve I, 1I, 1, p) − X i∈Ici(vI, 1I, 1, p)G 0(1 {i}) + cβ(vI, 1I, 1, p)α eHI) o weakly in`∞(∂B+1(0) × K) as n → ∞.
The proof of Theorem 10 is deferred to Appendix D.
REMARK11. Using the second part of Remark 7 and Equation (33) as well as the fact that
k1{i}◦ Θk = |Θi| ≤ maxj∈I|Θj| = k1I◦ Θk for all i ∈ I, we obtain that
Cov(G0(1{i}), eG(vI, 1I, p)) = E h v> 1I◦Θ k1I◦Θk p |Θi| i E[k1I◦ Θk] − E[|Θi|] E[k1I◦ Θk] ·E h v> 1I◦Θ k1I◦Θk p k1I◦ Θk i E[k1I◦ Θk] = CovΘIi, (vI>ΘI)p, and Cov( eHI, G0(1{i})) = E h − log1 ∧ |Θi| k1I◦Θk |Θi| i αE(k1I◦ Θk) = 1 αE− log Θ I i ΘIi ,
respectively, while Equation (26) implies that
Cov(G0(1{i}), G0(1{j})) = τ · E[Θi∧ Θj] = τI· E[ΘIi ∧ ΘIj]
for all i, j ∈ I. By Equation (34), eH(1I) is uncorrelated to all random variables eG(v, 1, 1I, p),
v ∈ ∂B1+(0), p ∈ K. Thus, we can calculate the asymptotic variance
e V (vI, p) = lim n→∞k · Var f Mn,k,I(vI, p) e Pn,k,I !
and obtain e V (vI, p) = 1 τI Var (vI>ΘI)p − 2X i∈I ci(vI, 1I, 1, p)Cov ΘIi, (vI>ΘI)p + τI· X i∈I X j∈I ci(vI, 1I, 1, p)cj(vI, 1I, 1, p)E[ΘIi ∧ ΘIj] − 2cβ(vI, 1I, 1, p) X i∈I ci(vI, 1I, 1, p)E[− log(ΘIi)ΘIi] + 1 τI cβ(vI, 1I, 1, p)2.
4. Application: Estimation of extremal coefficients. One of the most popular summary statistics for extremal dependence in the (sub-)vector (Xi∗)i∈I is the extremal coefficient τI
given by Equation (14), i.e.
τI= lim u→∞a ∗ (u)P max i∈I X ∗ i > u ∼P(maxi∈IX ∗ i > u) P(Xj∗> u) for all j ∈ I.
From this representation and the union bound for P(maxi∈IXi∗> u), it can be easily seen
that τI∈ [1, |I|] where τI= 1 corresponds to the case that all the components of (Xi)i∈I are
(asymptotically) fully dependent, while they are all asymptotically independent if τI= |I|.
This gives rise to the common interpretation of the extremal coefficient as the number of asymptotically independent variables among (Xi∗)i∈I.
From Equation (17), we obtain that
E h vI>ΘIi= Eh(vI>ΘI)+∧ 1 i = 1 τI
for all vI∈ ∂B1+(0). This motivates the use of the two classes of estimators considered in
Equation (21) and Equation (22), respectively. We will distinguish two cases; Section 4.1 collects the asymptotic normality results for the case of known marginal distributions where we can exploit the results for deterministic thresholds given in Section 3.1 with s = 1I and
β = 1. The results derived in Section 3.2 can be used when the margins are unknown as we will investigate in Section 4.2.
4.1. Known margins. In this section, we focus on the case where the margins are known, i.e. Xi∗ is observed.
4.1.1. The inefficiency of the benchmark. For this case, as a first class of estimators, we can directly consider the estimators based on relative frequencies of threshold exceedances by convex combinations, see Equation (21), namely
b Pn,u,I(conv)(vI) Pn,u,I = n −1Pn l=11{v > IX ∗ l > u} n−1Pn l=11{`I(Xl∗) > u} , vI∈ ∂B+1(0),
where we use the fact that v>IX∗> u already implies that `I(X∗) > u. This type of estimator
will serve as a benchmark. By standard arguments, for a sequence {un}n∈Nof thresholds such
that un→ ∞ and n/a∗(un) → ∞ as n → ∞, we obtain the asymptotic normality
r n a∗(u n) a∗(un) bP (conv) n,un,I(vI) − a ∗(u n)P(vI>X ∗> u n) a∗(un) bPn,un,I− a ∗(u n)P(`I(X∗) > un) ! d −→ N (0, Σ), where Σ =τIE[v > IΘI] τIE[vI>ΘI] τIE[v>IΘI] τI =1 1 1 τI .
Assuming that the bias is negligible, i.e. r n a∗(u n) a∗(un)P(v>IX ∗ > un) − 1 → 0 and (36) r n a∗(un) (a∗(un)P(`I(X∗) > un) − τI) → 0,
the Delta method yields asymptotic normality of the benchmark estimator for the case of known margins, 1 b τBK I,n (vI) =Pb (conv) n,un,I(vI) b Pn,un,I , v ∈ ∂B1+(0), i.e. r n a∗(un) 1 b τBK n (vI) − 1 τI d −→ N (0, τI−3(τI− 1)).
Here, it is important to note that the asymptotic variance does not depend on the choice of v ∈ ∂B1+(0).
Secondly, we consider moment-based estimators as discussed in Section 3.1, i.e. estima-tors based on empirical means of convex combinations, namely cMn,u,I(vI, 1)/ bPn,u,I with
weights vI∈ ∂B1+(0) as defined in Equation (22). Provided that the bias is negligible, i.e.
r n a∗(u n) a∗(un)E vI> X ∗ `I(X∗) · 1{`I(X∗) > un} − 1 → 0
and (36), we obtain the asymptotic normality
r n a∗(u n) c Mn,un,I(vI) b Pn,un,I − 1 τI ! d −→ N0, τI−1Var(vI>ΘI) ,
cf. the results in Corollary 6 and the first part of Remark 7.
Here, as v>IΘI≤ 1 a.s., we have that 1 τI Var(vI>ΘI) = 1 τI E[(vI>ΘI)2] − 1 τI2 ≤ 1 τI E[(vI>ΘI)] − 1 τI2 =τI− 1 τI3
and, therefore, the asymptotic variance of the moment-based approach is always less than or equal to the asymptotic variance of the benchmark estimator. Note that the variances of both estimators are equal, i.e. E[(vI>ΘI)2] = E[vI>ΘI] = 1/τI, if and only if ΘIi ≡ 1 a.s. for all i
with vi> 0. This is the case if and only if τI= 1 and then both asymptotic variances are equal
to 0. Thus, except for this degenerate case, the moment-based approach is always preferable to the benchmark in terms of the asymptotic variance.
Furthermore, in general, the asymptotic variance of the moment-based estimator depends on the vector vI ∈ ∂B1+(0). As we will see in Example 14, the minimal variance is typically
not achieved for a standard vector of the type v{i} for i ∈ I, i.e. the empirical estimators
of E[Θi], but by a non-trivial mixture of different basis vectors. This improvement can be
attributed to the fact that the moment-based estimator is linear in v. Thus optimizing in vI∈
∂B1+(0) can be interpreted as aggregating different estimators. It is well-known that the use of a combination of estimators can improve the accuracy. We note that we could also aggregate
the benchmark estimator for different v. However this strategy is much more intricate to study as the benchmark estimator is not linear in v.
Based on this discussion, for the remainder of Section 4.1, we focus on convex combina-tions of moment-based estimators rather than on the benchmark based on indicators. In the following, we optimize the asymptotic variance τI−1Var(vI>ΘI) w.r.t. the weight vectors vI
and we find some approximation of this best aggregation. We should notice that the optimal asymptotic variance Var(vI>ΘI) may be degenerate even when τI> 1. In these cases the
results hereinafter still hold with a null limit.
REMARK12. The asymptotic variance of the moment-based estimator is zero if and only
if there exists a weight vector vIsuch that vI>ΘII= 1/τIalmost surely. This happens in case
of an asymptotically independent vector X with v satisfying vi=|I|1 for i ∈ I. It might also
happen in more complex cases. For instance, assume that the vector ΘIItakes at only a finite number of different possible values θ(j)I , j ∈ 1, . . . , m. Then, vI is given as the solution of
the linear system vI>θ(j)I = 1/τI, j ∈ 1, . . . , m provided that the solution is in the simplex.
4.1.2. Variance minimization. We focus on the moment estimators cMn,u,I(vI, 1)/ bPn,u,Ifor
different vI∈ ∂B+1(0). As all of these estimators are consistent, it is appealing to choose the
estimator that provides the minimal asymptotic variance. Thus, in the following, we will focus on the determination of weight vectors v∗I∈ ∂B+
1 (0) that minimize the asymptotic variance
τI−1Var(vI>ΘII) = τI−3τI2· E[(v>IΘII)2] − 1
,
or, equivalently, the function
f(I): vI7→ E[(v>IΘII)2].
We will show that, under mild conditions, the minimizer of f(I)in ∂B1+(0) is unique.
PROPOSITION13. LetΘ be a spectral vector. Then, for any non-empty I ⊂ {1, . . . , d}, it
holds: If the matrixVI= (E[ΘIiΘIj])i,j∈I is conditionally positive definite, i.e.
a>VIa > 0
for all a ∈ R|I| \ {0} such that P
i∈Iai= 0, then the function vI 7→ f(I)(vI) attains its
minimum on∂B1+(0) at some unique vI∗∈ ∂B+1(0). PROOF. First, we note that
f(I)(vI) =
X
i∈I
X
j∈I
vivjE(ΘIiΘIj) = (vi)>i∈IVI(vi)i∈I.
By a slight abuse of notation, henceforth, we will write f(I)(vI) = vI>VIvI. Now, assume
that f(I)attains its minimum at two distinct points vI∈ ∂B1+(0) and wI∈ ∂B1+(0). Then,
for all λ ∈ (0, 1), we have λvI+ (1 − λ)wI∈ ∂B1+(0),Pi∈I(vi− wi) = 0 and, therefore,
f(I)(λvI+(1 − λ)wI) = (wI+ λ(vI− wI))>VI(vI+ (1 − λ)(wI− vI))
= − λ(1 − λ)(vI− wI)>VI(vI− wI) + λvI>VIvI+ (1 − λ)w>IVIwI
< λf(I)(vI) + (1 − λ)f(I)(wI),
EXAMPLE 14. In the case of a pairwise extremal coefficient τI with I = {i, j}, we obtain
an explicit expression for the minimizer v{i,j}∗ with v∗i, vj∗≥ 0 and v∗i+ vj∗= 1 of the function f(I)(v{i,j}) = E[(viΘIi+ vjΘIj)2] = (vi, vj)VI(vi, vj)>. To this end, we note that the function
vi7→ (vi, 1 − vi)VI
vi
1 − vi
= v2iE[(ΘIi)2] + 2vi(1 − vi)E[ΘiIΘIj] + (1 − vi)2E[(ΘIj)2]
is a quadratic function which possesses a unique minimizer vi∗ ∈ [0, 1] if and only if E[(ΘIi)2] − 2E[ΘIiΘIj] + E[(ΘIj)2] > 0, i.e. if and only if VI is conditionally positive
defi-nite, cf. Proposition 13. From
EΘIi · ΘIj = 1 E[`I(Θ)]· E ΘiΘj Θi∨ Θj = 1 E[`I(Θ)]· E [Θi ∧ Θj] = 1 E[`I(Θ)]· E [Θi + Θj− Θi∨ Θj] = τ τI · 2 τ − τI τ = 2 τI − 1, we obtain that (37) E[(ΘIi)2] ≥ [E(ΘIi)]2= 1 τ2 I ≥ 2 τI − 1 = E ΘI i · ΘIj and, analogously, (38) E[(ΘIj)2] ≥ EΘIi · ΘIj .
Note that equality in Equation (37) and in Equation (38) holds if and only if we have E[(ΘIi)2] = E[ΘIi] = 1 and E[(ΘIj)2] = E[ΘIj] = 1, respectively. Thus, for τI> 1, there exists
a unique minimizer which can be easily computed:
vi∗= E[(Θ
I
j)2] − E[ΘIiΘIj]
E[(ΘIi)2] − 2E[ΘIiΘIj] + E[(ΘIj)2]
=E[(Θ I j)2] − E[ΘIiΘIj] E[(ΘIi − ΘIj)2] . and v∗j = 1 − vi∗=E[(Θ I i)2] − E[ΘIiΘIj] E[(ΘIi − ΘIj)2] .
By Equations (37) and (38), we obtain v∗i, vj∗> 0, i.e. v{i,j}∗ ∈ ∂B1+(0). Some further calcu-lations lead to the minimal asymptotic variance τI−3(τI2· f(I)(v
I) − 1) with f(I)(vI∗) = E[(ΘIi)2] · E[(ΘIj)2] − E[ΘIiΘIj] 2 E[(ΘIi − ΘIj)2] .
4.1.3. Asymptotic normality of the plug-in estimator. We use the functional central limit theorems Corollary 6 and Theorem 10 to show that estimators with minimal asymptotic vari-ance can be obtained by plugging-in consistent estimators of the optimal weight vectors. The variance minimizing vector v∗Ican be estimated by taking the minimum of the empirical counterpart of f(I): b vn,u,I∗ = argminvI∈∂B+ 1(0) c Mn,u,I(vI, 2) b Pn,u,I .
It is important to note that, being constructed as the empirical second moment of a convex combination, vi7→ cMn,u,I(vI, 2)/ bPn,u,I is a positive semi-definite quadratic form. Thus, the
evaluation of cMn,u,I(vI, 2)/ bPn,u,I for a finite number of vectors vI is sufficient to
recon-struct the whole function { cMn,u,I(vI, 2)/ bPn,u,I : vI∈ ∂B1+(0)} and, thus, to calculate its
minimizer.
Here, even though f(I) is conditionally positive definite, i.e. the theoretical minimizer vI∗ is unique, under the assumptions of Proposition 13, in practice, definiteness of the empirical counterpart vI7→ cMn,u,I(vI, 2)/ bPn,u,Iis not guaranteed for a finite sample, i.e. its minimizer
b
vn,u,I∗ might no longer be unique. In such a case, we just choose an arbitrary minimizer. In particular,vb
∗
n,u,I can be an arbitrary vector of ∂B1+(0) if `I(Xl∗) ≤ u for all l ∈ {1, . . . , n}.
We first show consistency of this estimator.
PROPOSITION 15. Let Xl∗ = (Xl1∗, . . . , Xld∗)>, l ∈ N, be independent copies of a d-dimensional random vector X∗ satisfying the assumptions of Proposition 13 for all non-empty I ⊂ {1, . . . , d}. Furthermore, let {un}n∈N be a sequence such that un→ ∞ and
n/a∗(un) → ∞ as n → ∞ and assume that Equation (28) holds for p = 0, 2 and δ = 0.
Then,
b
vn,u∗ n,I−→ vp I∗ for any sequence of minimizersvbn,u∗
n,I and anyI ⊂ {1, . . . , d}. Furthermore, cMn,un,I(bv ∗ n,un,I, 2) →pf (I)(v∗ I).
PROOF. We observe that, for any non-empty I ⊂ {1, . . . , d}, we have f(I)(vI) = c(vI, 1I, 2)
for all vI∈ ∂B1+(0) and, thus, Corollary 6 yields
(39) supv I∈∂B+1(0)| cMn,un,I(vI, 2) − f (I)(v I)| p −→ 0.
As the continuous function f(I)attains its unique minimum on ∂B1+(0) at vI∗(see Proposition 13), for every ε > 0, there exists some η = η(ε) > 0 such that f(I)(v) − f(I)(v∗I) > η for all v with kv − v∗k > ε. Consequently, if kvb
∗
n,un,I− v
∗k > ε then we have that f(I)(
b v∗n,u
n,I) −
f(I)(v∗I) > η, which further implies f(I)(vbn,u∗ n,I) − cMn,un,I(vb
∗ n,un,I, 2) + cMn,un,I(v ∗ I, 2) − f(I)(vI∗) > η as cMn,un,I(vb ∗ n,un,I, 2) ≤ cMn,un,I(v ∗ I, 2) by definition ofvb ∗ n,un,I. In particular, sup v∈∂B+ 1(0) | cMn,un,I(v, 2) − f (I)(v)| > η/2. Noticing thatvb ∗
n,un,I is well defined only if
W 1≤l≤n`I(X ∗ l) > un, we obtain P(kbvn,u∗ n,I− v ∗ Ik > ε) ≤ P (`I(X∗) ≤ un)n+ P sup vI∈∂B+1(0) | cMn,un(vI, 2) − f (I)(v I)| > η 2 ,
which tends to 0 as n → ∞ by Equation (39). The statement that
c Mn,un,I(vb ∗ n,un,I, 2) →pf (I)(v∗ I)
follows from the facts that f(I)(vb
∗
n,un,I) − cMn,un,I(vb
∗
n,un,I, 2) →p 0 by Equation (39) and
f(I)( b v∗n,u n,I) − f (I)(v∗ I) →p0 due tovb ∗ n,un,I→pv ∗
I and the continuity of f(I).
The second part can be shown analogously using the results from Theorem 10 instead of Corollary 16.
Now, we are ready establish the desired limit theorem for the plug-in estimator, the moment-based estimator for the known margin case,
1 b τM K I,n =Mcn,un,I(vb ∗ n,un,I) b Pn,un,I ,
by combining the results of Proposition 15 with Corollary 6.
COROLLARY 16. Let Xl∗ = (Xl1∗, . . . , Xld∗)>, l ∈ N, be independent copies of a d-dimensional random vector X∗ satisfying the assumptions of Proposition 13 for all non-empty I ⊂ {1, . . . , d}. Furthermore, let {un}n∈N be a sequence such that un→ ∞ and
n/a∗(un) → ∞ as n → ∞ and assume that Equation (28) holds for p = 0, 1, 2 and δ = 0.
Then, for all non-emptyI ⊂ {1, . . . , d},
r n a∗(u n) 1 b τM K I,n − 1 τI ! d −→ N0, τI−1Var(vI∗>ΘI).
4.2. Unknown margins. Similarly, in case that the marginal distributions of X1, . . . , Xdare
unknown, i.e. Equation (4) is no longer satisfied and the index α of regular variation may be different from 1, we need to replace the deterministic threshold both in the benchmark esti-mator and in the moment-based estiesti-mator by a random one and include a Hill type estiesti-mator as described in Subsection 3.2.
4.2.1. The benchmark and the moment-based estimator. In case of the benchmark defined in Equation (21), the transformations above lead to the estimator
e Pn,k,I(conv)(vI) e Pn,k,I =n −1Pn l=11{vI>(Xl/Xk:n)αb> 1} n−1Pn l=11{`I(Xl/Xk:n) > 1} , vI∈ ∂B1+(0).
Of particular interest is the case that vI is a standard basis vector, i.e. vI = 1{i} for some
i ∈ I, since then the enumerator satifies ePn,k,I(conv)(1{i}) = k/n a.s., i.e. it possesses variance
zero. Thus, the estimator simplifies to 1 n kPen,k,I = 1 b Lk,n(1I)
where bLkn,nis the well-known non-parametric estimator for the stabe-tail dependence
func-tion [cf. Drees and Huang, 1998]. Provided that the bias is negligible and that the funcfunc-tion s 7→ E[ks ◦ ΘIk] is differentiable at s = 1I, this estimator is asymptotically normal:
p kn 1 b Lkn,n(1I) − 1 τI ! d −→ N 0,σ 2 L τI4
where the asymptotic variance of the stable dependence function estimator bLkn,n(1I) is
pro-vided in Chapter 7.4 of De Haan and Ferreira [2006] by the expression
σL2 = τI3X i∈I X j∈I ∂ ∂siE[ks ◦ Θ Ik] s=1I ∂ ∂sjE[ks ◦ Θ Ik] s=1I E[ΘIi ∧ ΘIj] − τI.
Using that E[ΘIi ∧ ΘIj] ≤ E(Θ∗i) = 1/τIfor all i ∈ I andPi∈I ∂s∂iE[ks ◦ Θ
Ik]
s=1I= 1 by
Euler’s homogeneous function theorem due to the positive 1-homogeneity of the function s 7→ E[ks ◦ ΘIk], we obtain that
σL2 τI4 ≤ τI2− τI τI4 = τI− 1 τI3 ,
i.e. the asymptotic variance of 1/bLkn,n(1I) is always smaller than the asymptotic variance
of the benchmark estimators for known margins, 1/τbI,n(vI). In particular, the estimator
be-comes more efficient by using order statistics even if the marginal distributions are known.
For the moment-based estimators, as already discussed in Subsection 3.2, the marginal trans-formations yield estimators of the type fMn,k,I(vI, 1)/ ePn,k,I, vI∈ ∂B+1(0). Provided that the
assumptions of Theorem 10 hold, we obtain the asymptotic normality
p kn f Mn,kn,I(vI) e Pn,kn,I − 1 τI ! d −→ N (0, eV (vI, 1)),
with eV (vI, 1) being given in Remark 11.
4.2.2. Variance minimization. Analogously to Section 4.1, we aim at choosing the estimator that provides the minimal asymptotic variance, i.e. use the weight vectorveI∈ ∂B
+
1(0) that
minimizes eV (·, 1), respectively. To this end, we first exploit the results given in Remark 11, to see that e V (vI, 1) = 1 τI VarvI>ΘI− 2X i∈I ci(vI, 1I, 1, 1)Cov ΘIi, v>IΘI + τI· X i∈I X j∈I ci(vI, 1I, 1, 1)cj(vI, 1I, 1, 1)E[ΘIi ∧ ΘIj] − 2cβ(vI, 1I, 1, 1) X i∈I ci(vI, 1I, 1, 1)E[− log(ΘIi)ΘIi] + 1 τI cβ(vI, 1I, 1, 1)2 = 1 τI Var vI>ΘI− τI X i∈Ici(vI, 1I, 1, 1)Θ I i + τI· X i∈I X j∈I ci(vI, 1I, 1, 1)cj(vI, 1I, 1, 1) E[ΘIi ∧ ΘIj] − E[ΘIi · ΘIj] − 2cβ(vI, 1I, 1, 1) X i∈I ci(vI, 1I, 1, 1)E[− log(ΘIi)ΘIi] + 1 τI cβ(vI, 1I, 1, 1)2, (40)
where we used that X i∈Ici(vI, 1I, 1, 1)E[Θ I i] 2 = 1 τI2 X i∈Ici(vI, 1I, 1, 1) 2 = 0,
which follows by Euler’s homogeneous function theorem as
(41) s 7→ c(vI, s, 1, 1) =
1 τI
vTs E[ks ◦ ΘIk]
is positively 0-homogeneous. By a straightforward computation, we obtain that e V (vI, 1) = 1 τI VarvI>ΘI− τIX i∈Ici(vI, 1I, 1, 1)Θ I i (42) + τI· (ci(vI, 1I, 1, 1))i∈I τI−1cβ(vI, 1I, 1, 1) > e VI (ci(vI, 1I, 1, 1))i∈I τI−1cβ(vI, 1I, 1, 1) with e VI=(E[Θ I
i ∧ ΘIj − ΘIiΘIj])i,j∈I(E[− log(ΘIi)ΘIi])i∈I
(E[− log(ΘIi)ΘIi])>i∈I 1
where both the expectation and the integral are interpreted componentwise. From the identity
e VI= E hZ 1 0 (1{ΘI i > u} − ΘIi)i∈I 1 + log(u) (1{ΘI i > u} − ΘIi)i∈I 1 + log(u) > du i ,
it can be seen that eV is always positive semi-definite. If it is even strictly positive definite, we obtain an analogous result to Proposition 13.
PROPOSITION17. LetΘ be a spectral vector. Assume that all the partial derivatives
ci(v, s, β, 1) = ∂ ∂si c(v, s, β, 1), i ∈ {1, . . . , d}, cβ(v, s, β, 1) = ∂ ∂βc(v, s, β, 1),
exist and are continuous on∂B1+(0) × A0δand that all the partial derivatives of the function s 7→ E(ks ◦ ΘIk)
exist and are continuous inAδ.
Then, for all non-emptyI ⊂ {1, . . . , d}, it holds: If the matrix eVI is strictly positive definite,
then the functionvI7→ eV (vI) attains its minimum on ∂B1+(0) at some uniqueveI∈ ∂B
+ 1(0).
PROOF. First, we note that all the derivatives ci, i ∈ {1, . . . , d}, and cβ are linear w.r.t. the
first component, i.e. they satisfy
ci(v, s, 1, 1) = Xd j=1vjci(1{j}, s, 1, 1), i ∈ I{1, . . . , d}, cβ(v, s, 1, 1) = Xd j=1vjcβ(1{j}, s, 1, 1)
for all v ∈ Rd. Thus, vI7→ eV (vI) is a quadratic form and, analogously to the proof of
Propo-sition 13, it remains to show that eV (wI) ≥ 0 for all wI∈ RdwithPi∈Iwi= 0 and equality
holds only if w = 0. We use the decomposition
e V (vI, 1) = g1(vI) + g2(vI) where g1(vI) = 1 τI Var v>IΘI− τI X i∈Ici(vI, 1I, 1, 1)Θ I i , g2(vI) = τI· (ci(vI, 1I, 1, 1))i∈I τI−1cβ(vI, 1I, 1, 1) > e VI (ci(vI, 1I, 1, 1))i∈I τI−1cβ(vI, 1I, 1, 1) ,
which follows directly from (42). By construction, g1(w) ≥ 0 as a variance. By the positive
definiteness of eVI, we have that g2(w) ≥ 0 for all w ∈ Rd, while equality holds only if
ci(wI, 1I, 1, 1) = 0 for all i ∈ I and cβ(wI, 1I, 1, 1) = 0. Differentiating (41) w.r.t. the i-th
component of s yields ci(wI, s, 1, 1) = 1 τI wi E[ks ◦ ΘIk] −(w Ts) · ∂ ∂siE[ks ◦ Θ Ik] E[ks ◦ ΘIk]2 ! .
In particular, for s = 1I and Pi∈Iwi = 0, we obtain ci(wI, 1I, 1, 1) = τI−1wi. Thus,
g2(w) = 0 only if wi= 0 for all i ∈ I, i.e. w = 0.
4.2.3. Asymptotic normality of the plug-in estimator. We note that, in contrast to f(I), there is no direct empirical counterpart of eV (·, 1). In order to enable the analytical minimization of that function, we will construct an estimator that retains the quadratic form of eV (·, 1). Noting that E[ΘIi] = τ
−1
I for all i ∈ I,
E[ΘIi ∧ ΘIj] = E[ΘIi] + E[ΘIj] − E[ΘIi ∨ ΘIj] =
2 τI −τ{i,j} τI , i, j ∈ I, and
E[− log(ΘIi)ΘIi] = cβ(1{i}, 1I, 1, 1), i ∈ I,
we can rewrite e V (vI, 1) = 1 τI VarvI>ΘI− 2X i∈I ci(vI, 1I, 1, 1)Cov ΘIi, vI>ΘI + τI· X i∈I X j∈I ci(vI, 1I, 1, 1)cj(vI, 1I, 1, 1)E[ΘIi ∧ ΘIj] − 2cβ(vI, 1I, 1, 1) X i∈I ci(vI, 1I, 1, 1)E[− log(ΘIi)ΘIi] + 1 τI cβ(vI, 1I, 1, 1)2 = 1 τI X i∈I X j∈I vivj E[ΘIiΘIj] − 1 τ2 I − 2X i∈I ci(vI, 1I, 1, 1) X j∈I vj E[ΘIiΘIj] − 1 τ2 I +X i∈I X j∈I ci(vI, 1I, 1, 1)cj(vI, 1I, 1, 1)(2 − τ{i,j}) − 2cβ(vI, 1I, 1, 1) X i∈I ci(vI, 1I, 1, 1)cβ(1{i}, 1I, 1, 1) + 1 τI cβ(vI, 1I, 1, 1)2. (43)
We will estimate thus eV (·, 1) replacing the unknown quantities τI, τ{i,j}, E[ΘIiΘIj],
ci(vI, 1I, 1, 1) and cβ(vI, 1I, 1, 1) in the right-hand side of (43) by consistent sequences
of estimatorseτI,n,eτ{i,j},n, eEn[Θ
I
iΘIj],eci,n(vI, 1I, 1, 1) andecβ,n(vI, 1I, 1, 1), respectively. If we define e ci,n(v, 1I, 1, 1) := Xd j=1vjeci,n(1{j}, 1I, 1, 1), i ∈ {1, . . . , d} and e ci,n(v, 1I, 1, 1) := Xd j=1vjecβ,n(1{j}, 1I, 1, 1)
as linear estimators and choose estimators eτ{i,j},n and eEn[Θ
I
iΘIj] that are symmetric in the
sense thatτe{i,j},n=τe{j,i},nand eEn[Θ
I
iΘIj] = eEn[ΘIjΘIi] for all i, j ∈ {1, . . . , d}, the resulting
variance estimator e gn,I(vI) = 1 e τI,n X i∈I X j∈I vivj Een[ΘIiΘIj] − 1 e τI,n2 ! − 2X i∈I eci,n(vI, 1I, 1, 1) X j∈I vj Een[ΘIiΘIj] − 1 e τI,n2 ! +X i∈I X j∈I e ci,n(vI, 1I, 1, 1)ecj,n(vI, 1I, 1, 1)(2 −eτ{i,j},n) − 2ecβ,n(vI, 1I, 1, 1) X i∈I e ci,n(vI, 1I, 1, 1)ecβ,n(1{i}, 1I, 1, 1) + 1 e τI,ne cβ,n(vI, 1I, 1, 1)2,
is of a quadratic form egn,I(vI) = v>IAenvI for a sequence of random symmetric matrices ( eAn)n∈N. Thus, a finite number of evaluations is sufficient to analytically calculate
e
vn,I= argminvI∈∂B1+(0)egn,I(vI).
Here, without further conditions on the underlying estimators, no kind of (semi-)definiteness is guaranteed for the matrices eAn. However, being a continuous function, vi7→egn,I(vI) attains its minimum on the compact set ∂B1+(0) at least once. Here, an arbitrary minimizer might be chosen forven,Iprovided that it is not unique.
Analogously to Proposition 15, consistency of the estimator can be shown.
PROPOSITION18. LetΘ be a spectral vector satisfying the assumptions of Proposition 17.
Then, for all non-emptyI ⊂ {1, . . . , d}, it holds: IfeτI,n→pτI,
e
En(ΘIiΘIj) →pE(ΘIiΘIj), i, j ∈ I,
e
ci,n(1{j}, 1I, 1, 1) →pci(1{j}, 1I, 1, 1), i, j ∈ I,
and ecβ,n(1{j}, 1I, 1, 1) →pcβ(1{j}, 1I, 1, 1), j ∈ I,
then we have thatven,I→pveI for any sequence of minimizersven,I. Furthermore,egn,I(ven,I) →pV (e veI).
PROOF. Recall that, by construction, egn,I(vI) = v
>
I AenvI for an appropriate sequence of random symmetric matrices ( eAn)n∈N, while eV (vI) = vI>Ave I for a conditionally positive definite matrix eA. The assumptions on the consistency of all the estimators involved in eAn
imply that k eAn− eAk∞→p0. Consequently,
sup
vI∈∂B1+(0)
|egn,I(vI) − eV (vI)| ≤ d2k eAn− eAk∞→p0,
which is the analogue to Equation (39). The remainder of the proof runs analogously to the proof of Proposition 15.
Combining the results of Proposition 18 and Theorem 10, we obtain the desired limit theorem for the plug-in estimator, the moment based estimator for the case of unknown margins
1 b τM U I,n =Mfn,k,I(ven,I, 1) e Pn,k,I .
COROLLARY 19. Let Xl, l ∈ N, be independent copies of a regularly varying [0, ∞)d
-valued random vector X satisfying Equation (2) such that the corresponding transformed vector X∗= (r1−1X1α, . . . , r−1d X1α), the spectral vector Θ and the sequence {kn}n∈N⊂ N
satisfy the assumptions of Proposition 18 for all non-emptyI ⊂ {1, . . . , d} and the assump-tions of Theorem 10 for K = {1}. Then, for all non-empty I ⊂ {1, . . . , d}, the plug-in esti-mator satisfies √ k 1 b τM U I,n − 1 τI ! d −→ N (0, eV (evI)) , n → ∞ .
In Appendix E we provide consistent estimators satisfying the conditions of Proposition 18. The quantities in egn,I(vI) based on moments of the spectral vector, i.e. τI, τ{i,j} and E[ΘIiΘIj], i, j ∈ I, can be directly estimated via the estimators constructed in Section 3.2. The
partial derivatives ci, i ∈ I, and cβ are approximated via difference quotients in Appendix E.
4.3. Simulated example. We consider the following bivariate max-linear model
X1= 1 2 + p + qmax{ Z1, pZ2, Z3, qZ4} X2= 1 2 + p + qmax{pZ1, Z2, qZ3, Z4}, where Z1, . . . , Z4∼i.i.d.Φ1with parameters p, q ∈ [0, 1].
By definition, the vector (X1, X2) possesses unit Fréchet marginal distributions with extremal
coefficient
τ = 4
2 + p + q,
i.e. the model can interpolate between full dependence (p = q = 1) and asymptotic indepen-dence (p = q = 0). The corresponding spectral vector Θ∗has a discrete distribution
Θ∗= (1, p) with probability 14, (p, 1) with probability 14, (1, q) with probability 14, (q, 1) with probability 14.
We compare the four types of estimators for τ , namely the benchmark estimators and the moment-based estimators, both with known and unknown margins as described in Sections 4.1 and 4.2, respectively. More precisely, in the case of known margins where we can work with standardized observations X1∗, X2∗, . . . , Xn∗ and use
• the benchmark estimator
1 b τBK I,n (v) = b Pn,u,{1,2}(conv) (v) Pn,u,{1,2} ;
• the moment-based plug-in estimator
1 b τI,nM K = c Mn,un,{1,2}(vb ∗ n,un,{1,2}) b Pn,un,{1,2} based on the estimated “optimal” weightsvb
∗
Note that, for simulated data from the bivariate max-linear model, no standardization is re-quired as the (unit Fréchet) marginal distributions are already identical and regularly varying with index α = 1. From Example 14, it can be seen that the variance minimizing linear com-bination is unique whenever p < 1 or q < 1 and is given by
v∗= 1 2, 1 2 .
As discussed in Section 4.1, the asymptotic distribution distribution of the benchmark esti-mator does not depend on the choice of vI∈ ∂B+1(0). Here we choosebτ
BK I,n =τb
BK I,n (v∗) as
benchmark.
Even though the margins are known, it might sometimes be beneficial to work with the rank transformations designed for the case of unknown margins as discussed in Section 4.2. Thus, we also consider
• the benchmark estimator
1 b τBU I,n = 1 n kPen,k,{1,2} = 1 b Lk,n(1)
taking the role of the benchmark; • the moment-based plug-in estimator
1 b τM U I,n =Mfn,k,{1,2}(ven,{1,2}, 1) e Pn,k,{1,2} based on estimated “optimal” weightsven,{1,2}.
From symmetry arguments, it can be easily seen that the “optimal” linear combination (if unique) is given by e v = 1 2, 1 2 .
We first compare these four estimators in terms of their asymptotic variances, that is, AVarBK = τI−3(τI− 1) (cf. Section 4.1), AVarM K = τI−1Var(v∗I
>Θ`
I) (cf. Corollary 16),
AVarBU= τI−4σ2L(cf. Section 4.2) and AVarM U = eV (veI) (cf. Corollary 19). The asymptotic standard deviations
√ AVarBK, √ AVarM K, √ AVarBU and √ AVarM U are displayed in Figure 1 as functions of p and q. First, it can be seen that all the estimators have asymptotic variance 0 in the case of full dependence (p = q = 1). While, in all the other cases, the benchmark type estimators have a strictly positive asymptotic variance, an asymptotic variance of 0 occurs for the new moment-based estimators also in the case of asymptotic independence and, if the margins are known, in all the cases where p = q as expected since Remark 12 applies in both cases. Complementing the results we obtained in Section 4 in a more general setting, Figure 1 provides a strict ordering of the four estimators in terms of the asymptotic variances:
AVarM K≤ AVarM U≤ AVarBU ≤ AVarBK for all 0 ≤ p, q ≤ 1,
where equalities hold in some degenerate cases with p, q ∈ {0, 1} only. Thus, in our bivariate max-linear model, we expect the new moment-based estimators to always outperform the benchmark estimators, but the differences are larger for the estimators with known margins than for their analogues based on order statistics.
Furthermore, we perform a simulation study to compare the estimators in a setting with finite sample sizes. Here, we consider three different scenarios:
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Asymptotic Std Dev Benchmark Estimator (Known Margins)
p q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Asymptotic Std Dev Moment Estimator (Known Margins)
p q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Asymptotic Std Dev Benchmark Estimator (Unknown Margins)
p q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Asymptotic Std Dev Moment Estimator (Unknown Margins)
p q −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
FIG 1. The asymptotic standard deviations of the four estimators, √ AVarBK, √ AVarM K, √ AVarBU and √
AVarM U(form top left to bottom right) as functions of the model parametersp and q.
• Scenario 1: p = 0.1, q = 0.2 (weak dependence) • Scenario 2: p = 0.4, q = 0.6 (medium dependence) • Scenario 3: p = 0.8, q = 0.9 (strong dependence)
In each of these scenarios, we consider n = 1 000 independent observations of X = (X1, X2)
and apply the estimators with a threshold un= Φ−11 (0.95) and k/n = 0.05, respectively.
For the estimation of the partial derivatives of c that are needed for the calculation of e
vn,{1,2}, we choose ε = k/n. We determine the biases and standard deviations by 5 000 repetitions. For comparison, we also calculate the theoretical standard deviations accord-ing to the central limit theorems, that is, we compute the square roots of the four terms n · P(X1> un) · AVarBK, n · P(X1> un) · AVarM K, k · AVarBU and k · AVarM U,
respec-tively.
The results are displayed in Table 1. First, it can be seen that, in the weak dependence sce-nario, all the biases are of a similar order as the standard deviations. In contrast, for the other two scenarios, compared to the standard deviation, the bias seems to be negligible for two of the four estimators, namelyτbI,nBK andbτI,nM U. The empirical standard deviations confirm the theoretical ordering of the four methods. For three of the four estimators, they are close to their theoretical counterparts. The only exception is the moment–based estimator in case of known margins,bτI,nM K, where the empirical standard deviation is significantly larger than the theoretical one. This can be explained that this estimator is in fact a plug-in estimator based on the estimated minimizervb
∗
n,un,{1,2} of the asymptotic variance. Even though the theoretical
