Asymptotically Unbiased Estimation of the Coefficient of Tail Dependence

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1 Asyptotically Unbiased Estiation of the Coefficient of Tail Dependence Yuri Goegebeur, Arelle Guillou To cite this version: Yuri Goegebeur, Arelle Guillou. Asyptotically Unbiased Estiation of the Coefficient of Tail Dependence. Scandinavian Journal of Statistics, Wiley, 2013, 40 (1), < /j x>. <hal > HAL Id: hal Subitted on 11 May 2016 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Asyptotically unbiased estiation of the coefficient of tail dependence YURI GOEGEBEUR Departent of Matheatics and Coputer Science, University of Southern Denark ARMELLE GUILLOU Institut Recherche Mathéatique Avancée, Université de Strasbourg et CNRS Abstract We introduce and study a class of weighted functional estiators for the coefficient of tail dependence in bivariate extree value statistics. Asyptotic norality of these estiators is established under a second order condition on the joint tail behavior, soe conditions on the weight function and for appropriately chosen sequences of interediate order statistics. Asyptotically unbiased estiators are constructed by judiciously chosen linear cobinations of weighted functional estiators, and variance optiality within this class of asyptotically unbiased estiators is discussed. The finite saple perforance of soe specific exaples fro our class of estiators and soe alternatives fro the recent literature are evaluated with a sall siulation experient. Keywords: bias-correction, coefficient of tail dependence, ultivariate extrees, second order condition. Running title: Unbiased estiation of tail dependence 1

3 1 Introduction In recent years, the area of ultivariate extree value theory has received a lot of attention, and any directions have been explored to infer on the characteristics of ultivariate extrees, including e.g. the estiation of indices or functions describing tail dependence and the estiation of the probability of extree failure sets. We refer to Ledford & Tawn (1996, 1997), Beirlant & Vandewalle (2002), Heffernan & Tawn (2004), Draisa et al. (2004), Peng (1999, 2010) to nae but a few. Focusing on the bivariate case where we have at our disposal pairs (X i,y i ), i = 1,...,n, being independent copies of the rando vector (X,Y) with joint distribution function F, it is atheatically convenient to assue that the arginals, F X and F Y, are known and that they are unit Fréchet distributions, i.e. F X (u) = F Y (u) = exp( 1/u) for u > 0. In this case, denoting by M X,n = ax 1 i n X i and by M Y,n = ax 1 i n Y i, it is well-known that, if there exists appropriate sequences a n and b n > 0 such that ( li P MX,n a n x, M Y,n a n n b n b n ) y = G(x,y) (1) where G is a non-degenerate distribution function, then G is called a bivariate extree value distribution. We say that M X,n and M Y,n are asyptotically independent if G(x,y) = G(x, )G(,y), for all x and y. This case is highly relevant in practice as for instance the bivariate noral distribution with correlation coefficient ρ < 1 (see our exaples in Section 4.1) has asyptotically independent argins. Unfortunately, if we want to estiate the probability that both X and Y exceed soe high thresholds, convergence (1) is not helpful. Indeed, noting that (1) can be reforulated in ters of the rando pair (X,Y) as ( MX,n a n P x, M ) Y,n a n y = F n (a n +b n x,a n +b n y), b n b n and taking the logarith, yields after soe rearrangeents that ( X li np an > x, Y a ) n > y = logg(x,y) logg(x, ) logg(,y), n b n b n 2

4 which is exactly 0 in the case of asyptotic independence. To solve this issue, Ledford & Tawn (1996, 1997) introduced a subodel assuing that the function t q(t) := P(1 F X (X) < t,1 F Y (Y) < t) is regularly varying at zero with index 1/η. This eans that q(t) = t 1/η l(t), where l is a function slowly varying at zero, i.e. an ultiately positive function satisfying l(tx)/l(t) 1 as t 0 for all x > 0 (see Bingha et al., 1987, p ). The paraeter η is called the coefficient of tail dependence, and satisfies η (0,1]. Ledford & Tawn (1996, 1997) otivate their odel by showing that any iportant and coonly used bivariate distribution functions can be written in this for; we also refer to Heffernan (2000) for an extensive list of exaples. The case η = 1 and li t 0 l(t) = c for soe 0 < c 1, corresponds to the asyptotic dependence case with degree c, whereas η (0,1) or η = 1 with li t 0 l(t) = 0 iplies asyptotic independence with degree 2η 1. Ledford & Tawn (1996) identify three types of asyptotic independence: if η (0,1/2), the pairs (X,Y) which exceed a sae high threshold occur less frequently than if X and Y are independent; η = 1/2: extrees of X and Y are close to independence and are exactly independent if l(.) = 1; η (1/2,1) or η = 1 and l(t) 0, the pairs (X,Y) which exceed a sae high threshold occur ore frequently than under exact independence. Thus the degree of asyptotic dependence depends on both η and l. Larger values of η indicate a stronger association between extree values of the two coponents X and Y. As we can iagine, different ethods have been proposed in the literature in order to estiate η fro observed data. In particular, as q is regularly varying with index 1/η, setting Z i = in{x i,y i }, we have P(Z i > z) = P(X i > z,y i > z) = z 1/η L(z) (2) with L(.) a slowly varying function at infinity. Thus the paraeter η can be viewed as the extree value index of the iniu of the two coponents, and hence all the classical type of estiators can be used, for instance the Hill (1975), oent (Dekkers et al., 1989) or axiu 3

5 likelihood (Sith, 1987) estiators. Unfortunately, as usual in the extree value context, the bias of all these estiators is a big challenge. In the univariate case, bias-reduced estiators have been introduced in e.g. Feuerverger & Hall (1999), Beirlant et al. (1999), and ore recently in Goes et al. (2008), to nae but a few, and such procedures have recently been considered in the bivariate case in Beirlant & Vandewalle (2002) using an exponential regression odel and in Beirlant et al. (2011) under a Hall-type condition on the slowly varying function L. The procedure by Beirlant & Vandewalle (2002) perfors quite well with respect to bias, though it should be noted that it is not necessarily asyptotically unbiased. The Beirlant et al. (2011) procedure on the other hand is asyptotically unbiased, though it does not take the uncertainty arising fro the arginal transforation by eans of the epirical distribution function into account. Both estiation ethods are axiu likelihood ethods and no explicit expressions for the estiators are available. In this paper we reconsider the bias-issue when estiating the coefficient of tail dependence η. In Section 2 we introduce a weighted functional estiator for η and establish its asyptotic norality under a second order condition on the joint tail behavior of the underlying distribution function, soe conditions on the weight function and for appropriately chosen sequences of interediate order statistics. A possible way to eliinate the asyptotic bias of such estiators, consisting in taking appropriate linear cobinations, is discussed in Section 3, where we also derive the iniu variance asyptotically unbiased estiator. Unlike the above entioned axiu likelihood approaches to bias-corrected estiation of η, our estiators have explicit expressions, aking the coputationally inexpensive. Soe siulations are discussed in Section 4, in which we copare the finite saple efficiency of soe exaples of weighted estiators with the likelihood based ethods of Beirlant & Vandewalle (2002) and Beirlant et al. (2011). The proofs of the results are postponed to the appendix, which is online available as a supporting inforation. 4

6 2 A functional estiator for η In this section we introduce a class of weighted functional estiators for the coefficient of tail dependence. Given a saple of independent and identically distributed (i.i.d.) rando vectors (X 1,Y 1 ),...,(X n,y n ), Ledford & Tawn (1996) proposed to standardize the arginal distributions to unit Fréchet argins, for instance by their epirical distribution function cobined with the inverse probability integral transfor, after which η can be estiated on the basis of the inia of the coponents by a classical estiator for the extree value index, e.g. the Hill estiator (Hill, 1975) or the oent estiator (Dekkers et al., 1989). This idea was also used in Beirlant & Vandewalle (2002), Draisa et al. (2004) and Beirlant et al. (2011), and will also for the basis for our ethodology. Forally, let R(X i ) denote the rank of X i aong (X 1,...,X n ) and R(Y i ) that of Y i aong (Y 1,...,Y n ), define { } 1 Z i = in log(r(x i )/(n+1)), 1, i = 1,...,n, (3) log(r(y i )/(n+1)) the iniu of unit Fréchet transfored argins, and denote by Z 1,n... Z n,n the corresponding ascending order statistics. Instead of assuing a transforation to unit Fréchet argins, we can alternatively work with unit Pareto argins, with distribution function given by F(u) = 1 1/u, u > 1. In that case (3) has to be replaced by { Z i = in Now, consider the functional 1 1 R(X i )/(n+1), 1 1 R(Y i )/(n+1) T K (z) := 1 0 log z(t) z(1) d(tk(t)), }, i = 1,...,n. for any easurable function z : [0,1] R (provided the right-hand side is defined and finite; otherwise T K (z) = 0), leading to the following class of estiators for η: ˆη (K) := T K (Q n ) = 1 0 log Q n(t) d(tk(t)) (4) Q n (1) 5

7 where Q n (t) := Z n t,n, 0 < t < n/, is the epirical quantile function, and K is a weight or kernel function with support on (0, 1). In order to establish the asyptotic distribution of (4) we have to introduce soe conditions on the kernel function K as well as a second order condition on the tail of the joint distribution function F. Concerning the weight function K we assue the following. Condition K: Let K(.) be a kernel such that (i) K(.) is continuously differentiable on (0, 1), (ii) 1 0 ( logt)d(tk(t)) = 1, (iii) there exists M > 0,0 r < 1/2 and p < 1 such that K(u) Mu r and K (u) Mu p r on (0,1). The conditions on K are not too restrictive and besides they are easy to verify for a given kernel function. For instance, our introduced class (4) includes the log weight-type estiator if K(u) = ( logu) ν /Γ(ν +1),u (0,1),ν 1, estiators based on the weight functions proposed in Goes et al. (2007) in the fraework of the estiation of the positive extree value index γ : K(u) = (1+ν)u ν and K(u) = (1+ν) 2 u ν ( logu), with u (0,1) and ν 0. Siilar conditions can be found in e.g. Gardes & Girard (2008a) and Goegebeur & Guillou (2011) in the fraework of the estiation of Weibull-type tails. We refer also to Mason (1981) for general results on asyptotic norality for linear cobinations of order statistics. For what concerns the second order tail behavior of F we use the following slightly odified version of the condition of Draisa et al. (2004) (see also de Haan & Stadtüller, 1996). This condition is not too restrictive and coonly used in estiation probles involving bivariate 6

8 extree values. Condition SO: Let (X,Y) be a rando vector with joint distribution function F and continuous arginal distribution functions F X and F Y such that ( li q 1 (t) 1 P(1 FX (X) < tx,1 F Y (Y) < ty) t 0 q(t) ) c(x,y) =: c 1 (x,y) (5) exists for all x 0,y 0 with x+y > 0, a positive function q and a function q 1 both tending to zero as t 0, and c 1 a function neither constant nor a ultiple of c. Moreover, we assue that the convergence is unifor on {(x,y) [0, ) 2 x 2 + y 2 = 1}, that c 1 is continuous and c 1 (x,x) = x 1/η (x τ 1)/τ. Reeber that q(t) := P(1 F X (X) < t,1 F Y (Y) < t). It can be shown that (5) iplies that q and q 1 are regularly varying at zero with index 1/η and τ 0, respectively. The function c is hoogeneous of order 1/η, that is c(tx,ty) = t 1/η c(x,y). In this paper, we only consider the case where η < 1 and τ > 0 (siilar restrictions were used in Beirlant & Vandewalle, 2002, and Beirlant et al., 2011). We refer to Section 4 for soe exaples of distributions satisfying SO. Our ain result is stated in Theore 1 below. It is essentially based on a odified version of Lea 6.2 in Draisa et al. (2004) by including a bias ter, cobined with the functional delta ethod. We refer to the supporting inforation for further details. Theore 1 Assue Condition K and the second order condition SO with a function c that is continuously differentiable. If such that /n 0 and q 1 (q 1 (/n)) λ, finite, there exists a standard Brownian otion W, such that (TK (Q n ) η) d η 1 0 t 1 W(t)d(tK(t)) ηw(1)k(1)+λ η τ 1 0 (t ητ 1)d(tK(t)). In particular (T K (Q n ) η) is asyptotically noral N (λab(k),av(k)) where AB(K) := η τ 1 AV(K) := η (t ητ 1)d(tK(t)), 1 0 in{s, t} st d(tk(t))d(sk(s)) η 2 K 2 (1). 7

9 We can now illustrate this theore in the special cases of the log weight-type kernels and the two weights proposed in Goes et al. (2007). This leads to the following corollaries. Corollary 1 (Log weight). Under the conditions of Theore 1, if K(u) = ( logu) ν /Γ(ν + 1),ν 1, then (TK (Q n ) η) ( d N λ η 2 (1+ητ) ν+1,η2γ(1+2ν) Γ 2 (1+ν) ). Corollary 2 Under the conditions of Theore 1, if K(u) = (1+ν)u ν,ν 0, then ( ) d (TK (Q n ) η) N λη 2 1+ν 1+ν +ητ,η2(1+ν)2. 1+2ν The Hill weight function K(u) = 1 is included in this faily (ν = 0). Corollary 3 Under the conditions of Theore 1, if K(u) = (1+ν) 2 u ν ( logu),ν 0, then ( ( ) ) d 1+ν 2 (TK (Q n ) η) N λη 2,2η 2 (1+ν)4 1+ν +ητ (1+2ν) 3. 3 Bias correction and estiation of the second order paraeter 3.1 An asyptotically unbiased estiator with iniu variance In this section we introduce a class of bias-corrected estiators for the paraeter η, and discuss variance optiality in the considered class. In particular, we propose to cobine two kernel-type estiators for η in order to cancel the asyptotic bias appearing in Theore 1. The eliination of bias by the construction of an appropriately weighted su of two estiators is also referred to as the generalized Jackknife ethodology; see Gray & Schucany (1972) for further details. More precisely, let K 1 (.) and K 2 (.) be two different kernels, both satisfying Condition K and let α be a real constant. Clearly K α (t) := αk 1 (t)+(1 α)k 2 (t) (6) satisfies also Condition K, and hence Theore 1 yields the asyptotic bias of this new estiator T Kα, naely AB(K α ) = η τ 1 0 (t ητ 1)d(tK α (t)) = αab(k 1 )+(1 α)ab(k 2 ). 8

10 Equating the right-hand side of the latter equation to 0 leads to the value of α that eliinates the asyptotic bias: α = AB(K 2 ) AB(K 2 ) AB(K 1 ), (7) provided AB(K 1 ) AB(K 2 ). This result is foralized in the following proposition. Proposition 1 Under the second order condition SO with a function c that is continuously differentiable, and assuing that K 1 and K 2 satisfy Condition K with AB(K 1 ) AB(K 2 ), we have that if such that /n 0 and q 1 (q 1 (/n)) λ, finite, then ( TKα (Q n ) η ) d N (0,AV(K α )). Thebias-correcting weight dependson η and τ, i.e. α = α (η,τ), which are unknownand hence need to be estiated fro the data. The following proposition states that replacing η and τ by initial consistent estiators η and τ, respectively, which possibly depend on sequences of upper order statistics different fro the one used for the bias-corrected estiator, does not change the liiting distribution of the noralized bias-corrected estiator. Denote ˆα := α ( η, τ). Proposition 2 Assue (i) the second order condition SO with a function c that is continuously differentiable, (ii) kernel functions K 1 and K 2 that satisfy Condition K with AB(K 1 ) AB(K 2 ) and that are such that α is continuously differentiable with respect to η and τ, and (iii) initial consistent estiators η and τ for η and τ, respectively. Then, if such that /n 0 and q 1 (q 1 (/n)) λ, finite, we have that ( TKˆα (Q n ) η ) d N (0,AV(K α )). We refer to the next subsection for an exaple of a consistent estiator for the second order paraeter τ. Now, inspired by Drees (1998a) and Proposition 3 in Gardes & Girard (2008b), we construct an asyptotically unbiased functional estiator with iniu variance. Theore 2 Let α opt := (1+ητ) 2 /(η 2 τ 2 ), K 1 (t) = 1 and K 2 (t) = (1+ητ)t ητ. Then K αopt (.) defined as in (6) is the asyptotically unbiased weight function with iniu variance aong unbiased weight functions satisfying Condition K. 9

11 We refer to the appendix for a proof of Theore 2. Also, direct coputations yield that the asyptotic variance of this optial asyptotically unbiased estiator is given by η 2 (1 + ητ) 2 /(ητ) 2. Corollary 4 Under the assuptions of Theore 1 and Theore 2, ( ) d T Kαopt (Q n ) η N ) (0,η 2(1+ητ)2 (ητ) 2. Fro Theore 1 we also obtain iediately the liiting distribution of the estiator for η that is obtained with K αopt in case one is-specifies the paraeter ητ at soe value. Let K a denote the kernel function K αopt with ητ fixed at the value a, i.e. K a (t) := (1+a) 2 /a 2 (1+a)(1+ 2a)t a /a 2, a > 0. Corollary 5 Under the second order condition SO with a function c that is continuously differentiable, we have that if such that /n 0 and q 1 (q 1 (/n)) λ, finite, then (TKa (Q n ) η) d N ) (λd(a,ητ),η 2(1+a)2 a 2, where d(a,ητ) := η 2 (1+a)(a ητ) a(1+ητ)(1+a+ητ). 3.2 Estiation of the second order paraeter τ In this section we introduce a siple estiator for the second order paraeter τ, the coefficient of regular variation of the function q 1. In the univariate Pareto-type fraework, the estiation of second order paraeters has been well studied and several good working estiators are available, we refer to Goegebeur et al. (2010) and the references therein for a recent account on the topic. In the ultivariate setting little work has been done in this respect. Peng (2010) introduced an estiator for a rate paraeter appearing in a second order condition that is slightly different fro condition SO, but apart fro this we are not aware of other attepts. 10

12 As the basic building block for our estiator we use the statistic n S(x,y) := 1{X i x,y i y}. i=1 Inspired by Goegebeur et al. (2010) and Peng (2010) we propose the following estiator for τ ˆτ k (x) := 1 log2 log H(x) H(2x) H(2x) H(4x), where H(x) := S( X n kx +1,n,Y n kx +1,n ) S ( X n k2x +1,n,Y n k2x +1,n ). The consistency of this estiator is established in the following proposition. Proposition 3 Assue that the second order condition SO is satisfied with a function c that has continuous first order partial derivatives. If k,n, suchthat k/n 0and nq(k/n)q 1 (k/n), then ˆτ k (x) P τ. 4 Siulation results The ai of this section is to illustrate the finite saple perforance of our bias-reduced estiator copared to alternative ethodologies. To this ai, we first study soe bivariate odels. Here we consider three distributions which satisfy our fraework and one, the bivariate noral case, which does not. We also shortly describe soe alternative estiation ethods fro the recent literature, and copare these with the ethodology presented in this paper in a sall siulation experient. 4.1 Soe bivariate distributions First note that, because of the assued continuous arginal distribution functions in condition SO we have that F X (X) and F Y (Y) are unifor (0,1) rando variables, and hence P(1 F X (X) < tx,1 F Y (Y) < ty) = tx+ty 1+C(1 tx,1 ty), where C is the copula function of the joint distribution function F. Verification of condition SO for a specific distribution can therefore be based copletely on the copula function C. 11

13 The Farlie Gubel Morgenstern distribution The Farlie Gubel Morgenstern copula function is given by C(u,v) = uv[1+ξ(1 u)(1 v)], (u,v) [0,1] 2, with ξ [ 1, 1]. Straightforward calculations lead to P(1 F X (X) < tx,1 F Y (Y) < ty) = t 2 xy [ 1+ξ ξt(x+y)+ξt 2 xy ]. In the case where ξ ( 1,1], P(1 F X (X) < tx,1 F Y (Y) < ty) P(1 F X (X) < t,1 F Y (Y) < t) [ = xy 1 ξt ] 1+ξ (x+y 2)+O( t 2), fro which one easily verifies that condition SO is satisfied with η = 0.5, c(x,y) = xy, c 1 (x,y) = xy(x+y 2)/2, q 1 (t) = 2ξt/(1+ξ), so τ = 1, while for the case ξ = 1 P(1 F X (X) < tx,1 F Y (Y) < ty) P(1 F X (X) < t,1 F Y (Y) < t) [ x+y = xy + t ] 2 4 (x+y 2xy)+O(t2 ), and hence condition SO is satisfied with η = 1/3, c(x,y) = xy(x + y)/2, c 1 (x,y) = xy(2xy x y)/2, q 1 (t) = t/2, so τ = 1. For the siulation we consider ξ = 1 and The Frank distribution The copula function for the bivariate Frank distribution is given by C(u,v) = 1 [ ξ log 1 (1 e ξu )(1 e ξv ] ) 1 e ξ, (u,v) [0,1] 2, where ξ > 0. Tedious coputations based on expansions of the above copula function lead to the following approxiation P(1 F X (X) < tx,1 F Y (Y) < ty) = [ ξ 1 e ξ t2 xy 1 ξt ] 2 (x+y)+o(t2 ), 12

14 fro which we deduce that P(1 F X (X) < tx,1 F Y (Y) < ty) P(1 F X (X) < t,1 F Y (Y) < t) [ = xy 1 ξt 2 (x+y 2)+O( t 2) ]. Hence condition SO is satisfied with η = 0.5, c(x,y) = xy, c 1 (x,y) = xy(x+y 2)/2, q 1 (t) = ξt and τ = 1. In the siulation we use ξ = 2 and 5. The Ali-Mikhail-Haq distribution For this distribution the copula function is given by C(u,v) = uv 1 ξ(1 u)(1 v), (u,v) [0,1]2, with ξ [ 1,1]. Fro the copula we easily establish that P(1 F X (X) < tx,1 F Y (Y) < ty) = t 2 xy [ 1+ξ ξt(x+y)+ξ(1+ξ)t 2 xy ξ 2 t 3 xy(x+y)+o(t 4 ) ]. First we consider the case where ξ ( 1,1]. Using Taylor s theore we obtain [ P(1 F X (X) < tx,1 F Y (Y) < ty) = xy 1 ξt ] P(1 F X (X) < t,1 F Y (Y) < t) 1+ξ (x+y 2)+O(t2 ), and hence condition SO is satisfied with η = 0.5, c(x,y) = xy, c 1 (x,y) = xy(x + y 2)/2, q 1 (t) = 2ξt/(1+ξ), so τ = 1. In case ξ = 1 we obtain [ ] P(1 F X (X) < tx,1 F Y (Y) < ty) x+y = xy t2 P(1 F X (X) < t,1 F Y (Y) < t) 2 2 (x+y)(xy 1)+O(t3 ), leading toη = 1/3, c(x,y) = xy(x+y)/2, c 1 (x,y) = xy(x+y)(xy 1)/4, q 1 (t) = 2t 2, and τ = 2. We will consider ξ = 1. The bivariate noral distribution 13

15 The bivariate noral distribution with ean 0, variance 1 and correlation coefficient ρ ( 1, 1) satisfies condition SO with η = (1 + ρ)/2. We refer to Draisa et al. (2004) and Ledford & Tawn (1997) for further details. Unfortunately, it falls outside the scope of the present paper since τ = 0. However, we add this distribution in our siulation study in order to exaine the robustness of our approach. We consider the cases ρ = 0.5 and ρ = 0.5, for which η = 0.25 and η = 0.75, respectively. 4.2 Estiators In the siulation experient we copare the optial (in the sense of inial asyptotic variance) bias-corrected estiator with the Hill estiator and two estiators fro the recent extree value literature, described below. Note that the optial weight in the bias-corrected estiator involves the paraeter ητ. In the siulation experient, for all distributions, except the bivariate noral, we will explore two strategies naely fixing ητ at its true value and a is-specification of this paraeter at ητ = 1. For the bivariate noral distribution, which has τ = 0, we consider fixing ητ at 0.5 and 1. Fixing second (or higher) order distributional paraeters at a canonical choice, e.g. taking ητ = 1, is a coon approach in extree value statistics. For instance, in the univariate extree value fraework, the second order paraeter ρ is often fixed at -1 in practical ipleentations of bias-corrected estiators for the extree value index; see e.g. Feuerverger & Hall (1999) and Goes & Martins (2004). Alternatively, the second order paraeter τ could also be estiated with the estiator proposed in Section 3.2. Though this estiator is proven to be consistent, its practical use needs further investigation, and this will be pursued in future research on the estiation of the second order paraeter in the ultivariate extree value fraework. Beirlant & Vandewalle (2002) proposed to estiate η by applying the axiu likelihood ethod to the following approxiate exponential regression odel for scaled log-ratios ( ) Zn j+1,n Z n,n η jlog Z n j,n Z n,n 1 (j/) η f j, j = 1,..., 1, where f 1,...,f 1 are i.i.d. standard exponential rando variables. The resulting estiator will herebedenoted as ˆη BV. Beirlant & Vandewalle (2002) proved that ˆη BV, whenappropriately 14

16 noralized, is asyptotically norally distributed, though it should be entioned that it is not asyptotically unbiased. Nevertheless, in the siulations reported in their paper, the estiator perfors quite well with respect to bias. Beirlant et al. (2011) introduced the extended Pareto distribution as approxiate odel for relative excesses over a threshold P(Y w > tz) P(Y w > t) [z(1+δ w(t) δ w (t)z τ )] 1/η, where Y w := in(z 1,Z 2 w/(1 w)), Z 1 and Z 2 unit Fréchet rando variables, w a tuning paraeter, and obtained an estiator for η, denoted here as ˆη B, fro linearized score functions. We refer to Beirlant et al. (2011) for ore details about w and δ w (t). The estiator was proven to be asyptotically unbiased with inial asyptotic variance η 2 (1+ητ) 2 /(ητ) 2, though the uncertainty arising fro the arginal transforations by eans of the epirical distribution functions was not explicitly taken into account. The paraeter w was introduced to estiate probabilities in joint tail regions, but has little practical relevance for the estiation of η, and therefore we fix it at 0.5. Concerning the second order paraeter τ, Beirlant et al. (2011) showed that replacing it by a consistent estiator does not change the liiting distribution of ˆη B, though no estiator for that paraeter was proposed. As suggested by these authors, in the siulation experient we ipleent their estiator with τ fixed at the canonical choice 1. In Figure 1 we show the asyptotic standard deviations of the estiators under study as a function of η. Siilar to the univariate case, the Hill estiator, which will be denoted by ˆη H has the sallest possible variance, but it is not unbiased. The estiators T Kαopt and ˆη B have the sae asyptotic variance, which is here shown for three values of τ naely τ = 0.5, 1, 2. Fro Corollary 4, the increase in variance relative to the Hill estiator is iediately clear, and is given by (1+ητ) 2 /(ητ) 2. The Beirlant & Vandewalle (2002) estiator ˆη BV has approxiately thesaevariance ast Kαopt and ˆη B incase τ = 1. For values of τ below 1, ˆη BV will typically have a saller asyptotic variance than T Kαopt and ˆη B, though it is not asyptotically unbiased. Insert Figure 1 here 15

17 4.3 Results The siulation experient considers the distributions listed in Section 4.1, with both unit Pareto and unit Fréchet argins. For each of the distributions we generated 1000 saples of size n = 500, and coputed all the above entioned estiators for = 5,...,499, where denotes the nuber of upper order statistics of the Z i observations used in the estiation of η. In Figures 2 till 6 we show the saple ean (left) and the epirical ean squared error (MSE) (right) as a function of for the estiators T Kαopt with the true value of ητ (black solid line), T Kαopt with ητ = 1 (black dashed line), ˆη H (black dotted line), ˆη B (grey solid line) and ˆη BV (grey dashed line). For the bivariate noral copula function the results are shown in Figures 7 and 8. Here T Kαopt is coputed with ητ = 0.5 and ητ = 1. Fro the siulation results we can draw the following conclusions: (i) The Hill estiator is generally biased, though the bias sees to be a ore severe proble for unit Fréchet arginal distributions than for unit Pareto arginal distributions. This observation is in agreeent with the theoretical considerations in Drees (1998a, b). For the distributions where the bias is a proble, like e.g. the Farlie Gubel Morgenstern with unit Fréchet argins and the distributions based on Frank s copula, the estiators T Kαopt, ˆη B and ˆη BV typically outperfor ˆη H in ters of inial MSE. Also, the saple paths of these estiators show a longer stable part copared to ˆη H. (ii) The estiators ˆη B and ˆη BV exhibit a very siilar behavior, both in ters of bias and MSE, but ˆη B behavessoewhatlesserraticthan ˆη BV, ascanbeseenfroitsorestablesaplepaths. (iii) As expected, no estiator perfors uniforly best, but T Kαopt is highly copetitive copared to ˆη B and ˆη BV, especially if one takes into account that, unlike the axiu likelihood based estiators, it can be coputed directly fro the data without needing an iterative optiization schee. Also, for T Kαopt, is-specifying ητ at the canonical choice 1, often turns out to work better in practice than using the true value of that paraeter. This can be explained fro Corollary 5, at least for the cases with τ > 0. Indeed, fro Corollary 5 the asyptotic 16

18 variance is decreasing in the paraeter a, and hence the estiator with ητ fixed at the value 1 will have a saller variance copared to the estiator obtained with the true value of ητ (for the distributions under consideration in the siulation ητ < 1). This difference in variance can be considerable, e.g. in case η = 1/3 and τ = 1 we have an asyptotic variance of 1.78 in case one uses the true value of ητ, copared to an asyptotic variance of only 0.44 when one fixes ητ at 1. Using the true value of ητ of course eliinates the bias, at least theoretically. However, in the siulations fixing ητ at 1 perfors often as well as using the true value of ητ in ters of bias, and in soe cases even better, which then lead to the better perforance in ters of iniu MSE. Insert Figures 2-8 here 5 Conclusion In this paper we considered the estiation of the coefficient of tail dependence in bivariate extree value statistics, and introduced a class of weighted functional estiators for this paraeter. By using the functional delta ethod the estiators were shown to be asyptotically noral under a second order condition on the joint tail behavior, soe conditions on the kernel function and for an interediate sequence of upper order statistics. By taking appropriately chosen weighted sus of two (biased) estiators we obtained a class of asyptotically unbiased estiators, and we established variance optiality within this class. The siulations indicated that the estiator is highly copetitive with recent alternatives fro the extree value literature, especially if one takes their low coputational deands into account. In future work we will focus on the further developent of estiators for the second order paraeter τ, as well as their practical ipleentation. Acknowledgeent The authors are very grateful to the associate editor and the two referees for their careful reading of the paper and their coents that lead to significant iproveents of the initial 17

19 draft. Yuri Goegebeur s research was supported by a grant fro the Danish Natural Science Research Council. Supporting inforation Additional inforation for this article is available online including: a suppleentary appendix containing detailed proofs of Theore 1, Proposition 2, Theore 2, Proposition 3. References [1] Beirlant, J., Dierckx, G., Goegebeur, Y. & Matthys, G. (1999). Tail index estiation and an exponential regression odel. Extrees 2, [2] Beirlant, J., Dierckx, G. & Guillou, A. (2011). Bias-reduced estiators for bivariate tail odelling. Insurance Math. Econo. 49, [3] Beirlant, J. & Vandewalle, B. (2002). Soe coents on the estiation of a dependence index in bivariate extree value in statistics. Statist. Probab. Lett. 60, [4] Bingha, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular variation. Cabridge University Press, Cabridge. [5] Dekkers, A.L.M., Einahl, J.H.J. & de Haan, L. (1989). A oent estiator for the index of an extree-value distribution. Ann. Statist. 17, [6] Draisa, G., Drees, H., Ferreira, A. & de Haan, L. (2004). Bivariate tail estiation: dependence in asyptotic independence. Bernoulli 10, [7] Drees, H. (1998a). A general class of estiators of the extree value index. J. Statist. Plann. Inference 66, [8] Drees, H. (1998b). On sooth statistical tail functionals. Scand. J. Stat. 25, [9] Feuerverger, A. & Hall, P. (1999). Estiating a tail exponent by odelling departure fro a Pareto distribution. Ann. Statist. 27,

20 [10] Gardes, L. & Girard, S. (2008a). Estiation of the Weibull tail-coefficient with linear cobination of upper order statistics. J. Statist. Plann. Inference 138, [11] Gardes, L. & Girard, S. (2008b). A oving window approach for non paraetric estiation of the conditional tail index. J. Multivariate Anal. 99, [12] Goegebeur, Y., Beirlant, J. & de Wet, T. (2010). Kernel estiators for the second order paraeter in extree value statistics. J. Statist. Plann. Inference 140, [13] Goegebeur, Y. & Guillou, A. (2011). A weighted ean excess function approach to the estiation of Weibull-type tails. TEST 20, [14] Goes, M.I., de Haan, L. & Rodrigues, L.H. (2008). Tail index estiation for heavy-tailed odels: accoodation of bias in weighted log-excesses. J. R. Stat. Soc. Ser. B Stat. Methodol. 70, [15] Goes, M.I.& Martins, M.J.(2004). Bias reduction and explicit sei-paraetric estiation of the tail index. J. Statist. Plann. Inference 124, [16] Goes, M.I., Miranda, C. & Viseu, C. (2007). Reduced bias tail index estiation and the Jackknife ethodology. Stat. Neerl. 61, [17] Gray, H.L. & Schucany, W.R. (1972). The generalized Jackknife statistic. Marcel Dekker. [18] de Haan, L. & Stadtüller, U. (1996). Generalized regular variation of second order. J. Aust. Math. Soc. 61, [19] Heffernan, J. E. (2000). A directory of coefficients of tail dependence. Extrees 3, [20] Heffernan, J. E. & Tawn, J. A. (2004). A conditional approach for ultivariate extree values. J. R. Stat. Soc. Ser. B Stat. Methodol. 66, [21] Hill, B.M. (1975). A siple general approach to inference about the tail of a distribution. Ann. Statist. 3, [22] Ledford, A.W. & Tawn, J.A. (1996). Statistics for near independence in ultivariate extree values. Bioetrika 83,

21 [23] Ledford, A.W. & Tawn, J.A. (1997). Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B Stat. Methodol. 59, [24] Mason, D.M. (1981). Asyptotic norality of linear cobinations of order statistics with a sooth score function. Ann. Statist. 9, [25] Peng, L. (1999). Estiation of the coefficient of tail dependence in bivariate extrees. Statist. Probab. Lett. 43, [26] Peng, L. (2010). A practical way for estiating tail dependence functions. Statist. Sinica 20, [27] Sith, R.L. (1987). Estiating tails of probability distributions. Ann. Statist. 15, Corresponding author: Yuri Goegebeur, Departent of Matheatics and Coputer Science, University of Southern Denark, Capusvej 55, 5230 Odense M, Denark. E-ail: yuri.goegebeur@stat.sdu.dk. 20

22 Astd eta Figure 1: Asyptotic standard deviations as a function of η: T Kαopt and ˆη B when τ = 1 (black solid line), τ = 2 (black dashed line) and τ = 0.5 (black dotted line), ˆη H (grey solid line), and ˆη BV (grey dashed line) estiators. 21

23 eta MSE eta MSE Figure 2: Bivariate Farlie Gubel Morgenstern copula with ξ = 1 (row 1) and ξ = 0.25 (row 2), saple ean (left) and MSE (right), transforation to unit Pareto argins: Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and the true value of ητ (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 22

24 eta MSE eta MSE k Figure 3: Bivariate Farlie Gubel Morgenstern copula with ξ = 1 (row 1) and ξ = 0.25 (row 2), saple ean (left) and MSE (right), transforation to unit Fréchet argins: Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and the true value of ητ (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey, solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 23

25 eta MSE eta MSE Figure 4: Bivariate Frank copula with ξ = 2 (row 1) and ξ = 5 (row 2), saple ean (left) and MSE (right), transforation to unit Pareto argins: Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and ητ = 0.5 (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey, solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 24

26 eta MSE eta MSE Figure 5: Bivariate Frank copula with ξ = 2 (row 1) and ξ = 5 (row 2), saple ean (left) and MSE (right), transforation to unit Fréchet argins: Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and ητ = 0.5 (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey, solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 25

27 eta MSE eta MSE Figure 6: Bivariate Ali-Mikhail-Haq copula with ξ = 1, unit Pareto argins (row 1) and unit Fréchet argins (row 2), saple ean (left) and MSE (right): Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and ητ = 2/3 (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey, solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 26

28 eta MSE eta MSE Figure 7: Bivariate noral copula with µ 1 = µ 2 = 0, σ1 2 = σ2 2 = 1, ρ = 0.5 (row 1) and ρ = 0.5 (row 2), saple ean (left) and MSE (right), transforation to unit Pareto argins: Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and ητ = 0.5 (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey, solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 27

29 eta MSE eta MSE Figure 8: Bivariate noral copula with µ 1 = µ 2 = 0, σ1 2 = σ2 2 = 1, ρ = 0.5 (row 1) and ρ = 0.5 (row 2), saple ean (left) and MSE (right), transforation to unit Fréchet argins: Hill (black, dotted), T K (Q n ) with optial bias-correcting weight and ητ = 0.5 (black, solid), T K (Q n ) with optial bias-correcting weight and ητ = 1 (black, dashed), ˆη B (grey, solid) and ˆη BV (grey, dashed) as a function of. The horizontal line is the true value of η. 28