Klüppelberg, Kuhn, Peng: Estimating Tail Dependence of Elliptical Distributions

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1 Klüppelberg, Kuhn, Peng: Estimating Tail Dependence of liptical Distributions Sonderforschungsbereich 386, Paper Online unter: Projektpartner

2 Estimating Tail Dependence of liptical Distributions By CLAUDIA KLÜPPELBERG, GABRIEL KUHN Center for Mathematical Sciences, Munich University of Technology, D Garching, Germany. and LIANG PENG School of Mathematics, Georgia Institute of Technology, Atlanta, GA , USA. Summary Recently there has been an increasing interest in applying elliptical distributions to risk management. Under weak conditions, Hult and Lindskog 22 showed that a random vector with an elliptical distribution is in the domain of attraction of a multivariate extreme value distribution. In this chapter we study two estimators for the tail dependence function, which are based on extreme value theory and the structure of an elliptical distribution, respectively. After deriving second order regular variation estimates and proving asymptotic normality for both estimators, we show that the estimator based on the structure of an elliptical distribution is better than that based on extreme value theory in terms of both asymptotic variance and optimal asymptotic mean squared error. Our theoretical results are confirmed by a simulation study. Keywords: asymptotic normality, elliptical distribution, regular variation, tail copula, tail dependence function. Introduction Let X,Y, X,Y, X 2,Y 2, be independent random vectors with common distribution function F and continuous marginals F X and F Y. Define the tail copula λx, y := t t P F XX tx, F Y Y ty

3 for x,y. Then λ, is called the upper tail dependence coefficient, see e.g. Joe 997 and, by Huang 992, lx,y := x + y λx,y is called the tail dependence function. Assuming that X,Y is in the domain of attraction of a bivariate extreme value distribution, there exist several estimators for estimating the tail dependence function lx, y, see Huang 992, Einmahl, de Haan and Huang 993 and de Haan and Resnick 993. The optimal rate of convergence for estimating lx, y is given by Drees and Huang 998. An alternative method for estimating lx, y is via estimating the spectral measure, see Einmahl, de Haan and Sinha 997 and Einmahl, de Haan and Piterbarg 2. For modeling dependence of extremes parametrically, we refer to Tawn 988 and Ledford and Tawn 997. Triggered by financial risk management problems we observe an increasing interest in elliptical distributions as natural extensions of the normal family allowing for the modeling of heavy tails and extreme dependence. The vector X,Y is elliptically distributed, if X,Y T = µ + GAU 2,. where µ = µ X,µ Y T, G > is a random variable, called generating variable, A R 2 2 is a deterministic matrix with AA T =: Σ := σ 2 ρσυ ρσυ υ 2 and rankσ = 2, U 2 is a 2-dimensional random vector uniformly distributed on the unit hyper-sphere S 2 := {z R 2 : z = }, and U 2 is independent of G. Note that ρ is termed as the linear correlation coefficient of Σ. Under certain conditions, Hult and Lindskog 22 showed that regular variation of G with index α >, i.e., t Gtx/ Gt = x α for all x >, notation: G R α is equivalent to regular variation of X,Y with the same index α > see Resnick 987 for the definition of multivariate regular variation. Moreover, if G R α, then π/2 π/2 λ, = cos φ α dφ / cos φ α dφ..2 π/2 arcsin ρ/2 Here we are interested in estimating the dependence function λx, y by assuming that G RV α for some α >. Since G RV α implies that 2

4 X,Y is in the domain of attraction of an extreme value distribution, a naive estimator is to apply Huang s estimator by ignoring the structure of the elliptical distribution, i.e., Hu k Hu,nx,y := k Hu n I X i X n xkhu,n, Y i Y n ykhu,n,.3 i= where X,n X n,n and Y,n Y n,n denote the order statistics of X,...,X n and Y,...,Y n, respectively, k Hu = k Hu n n and k Hu /n n. The same estimator has been analysed by Schmidt and Stadtmüller 25; see their equation 4.4. The aim of this chapter is two-fold. Firstly, we suggest a new estimator, which exploits the structure of an elliptical distribution similar to.2. Secondly, we aim at determining the optimal number of order statistics to be used in both estimators. The choice will be based on the asymptotic mean squared error of the estimators. Our chapter is organized as follows. We first derive an expression for λx,y, which generalizes equation.2, and then construct a new estimator for λx,y via this expression; see section 2 for details. After deriving the second order behavior for elliptical distributions and the iting distributions of both estimators in section 2, we show that the new estimator is better than the naive empirical estimator from Huang in terms of both asymptotic variance and optimal asymptotic mean squared error in section 3. More importantly, the optimal choice of the sample fraction for the new estimator is the same as that for Hill s estimator Hill 975. That is, all data-driven methods for choosing the optimal sample fraction for Hill s estimator can be applied to our new estimator directly. A simulation study is provided in section 3 as well. All proofs are summarized in section 4. 2 Methodology and Main Results The following theorem gives an expression for λx, y, which will be employed to construct an estimator. Theorem 2.. Suppose X, Y defined in. holds with σ >, υ >, ρ < and G R α for some α >. Further, define gt := arctan t ρ/ ρ 2 [ arcsin ρ,π/2], t R. 3

5 Then λx, y = π/2 π/2 cos φ α dφ /2 gx/y /α xcosφα dφ gx/y /α + y sinφ + arcsinρ α dφ. arcsin ρ Hu In order to derive the asymptotic normality of k Hu,nx,y, it is known that a second order condition is needed. Here we seek the relation of the second order behavior among the tail copula λx,y, X 2 + Y 2 and G; see the next two theorems for details. In the setting of. assume that there exists At such that for all x > and some β PG tx/pg t x α t At = x αxβ, 2. β where β is called a second order regular variation parameter, see de Haan and Stadtmüller 996. Additionally, we assume t t2 At =: a [, ]. 2.2 Since A R β, a = for β < 2 and a = for β 2, ]. The following two theorems derive the corresponding second order condition for X 2 + Y 2 and the tail copula λx,y. Theorem 2.2. Assume that the conditions of Theorem 2., 2. and 2.2 hold. Further, define Then, for all x >, d φ = σ 2 cos 2 φ + υ 2 sin 2 φ + arcsin ρ, d 2 φ = µ X σ cos φ + µ Y υ sinφ + arcsin ρ. P{ X 2 + Y 2 tx}/p{ X 2 + Y 2 t} x α t t 2 + At π { a = x α d φ α/2 x β dφ + a β + α x 2 π d φ α/2 + a 2 π d φ α β/2 dφ [ αd 2 φ 2 + d φµ 2 X + µ 2 Y ] dφ }

6 Also, for all x > and V x := inf{y : P X 2 + Y 2 > y x }, t V tx/v t x /α FY t 2 + AFY t π π = x /α d φ α/2 dφ d φ α/2 2 β /α dφ υ α π sin φα dφ { a x β/α π d φ α β/2 dφ + a αβ + x 2/α π d φ α/2 + a 2 [ αd 2 φ 2 + d φµ 2 X + µ 2 Y ] dφ } =: x /α B 2.4 x. 2.4 Especially, when µ X = µ Y =, we have for all x > V tx/v t x /α t AFY t = x /αxβ/α υ β αβ π π d φ α β/2 dφ d φ α/2 dφ π sin φα dφ π d φ dφ α/2 β/α =: x /α B 2.5 x. 2.5 Theorem 2.3. Assume that the conditions of Theorem 2. and 2. hold. Further, define S 2 + := { z R 2 : z and z = } and B 2.6 x := x x β/α π π sin φ α d φ sin φ α β dφ 2.6. β 5

7 Then, t P F X X tx, F Y Y ty λx,y t A FY t { = υ β x π/2 [ x β/α cos φ α β cos φ α] dφ β gx/y /α + y β gx/y /α arcsin ρ [ y β/α sinφ+arcsinρ α β sinφ+arcsinρ α] dφ π/2 +B 2.6 x cosφ α dφ gx/y /α +B 2.6 y λx,y β gx/y /α arcsin ρ π/2 /2 sinφ + arcsin ρ α dφ cosφ α cos φ β } π/2 dφ cos φ α dφ /2 =: B 2.7 x,y holds for all x,y and uniformly on S Now we are ready to define our new estimator. Put Z i = Xi 2 + Y i 2 for i =,...,n and let Z,n Z n,n denote their order statistics. First we estimate the index α by Hill s estimator, which is defined as α H k,n := k k i= log Z n i+,n log Z n k,n, where k = k n and k /n as n. Now let X,Y and X,Ỹ be iid with elliptical distribution. Then, it follows from Hult and Lindskog 22 that τ = 2/π arcsin ρ, where τ is called Kendall s tau and defined by τ := P X X Y Ỹ > P X X Y Ỹ <. As usual, we estimate Kendall s tau by τ n := 2 nn i,j n sign X i X j Y i Y j, 6

8 which results in estimating ρ by π ρ n = sin 2 τ n. Hence, we can estimate λx,y by replacing ρ and α in Theorem 2. by ρ n and α k H,n, respectively. Let us denote this estimator by We remark that k,n k,nx,y. 2.8, was mentioned by Schmidt 23, but without further Hu k Hu,nx,y and study. The following theorem shows the asymptotic normalities of k,n x,y, which allows us to compare these two estimators theoretically. Theorem 2.4. Assume that the conditions of Theorem 2. and 2. hold. Suppose k Hu = k Hu n n, k Hu /n n and khu AFY n k Hu /n : K Hu, for K Hu <. Then, as n, sup x,y T = o p, k Hu Hu k Hu,nx,y λx,y for any T >, where B 2.7 x,y is defined in Theorem 2.3, Bx,y = Wx,y λx,y Wx, x K Hu B 2.7 x,y Bx,y 2.9 λx,y y W,y, and Wx,y is a Wiener process with zero mean and covariance structure E Wx,y Wx 2,y 2 = x x 2 + y y 2 λx x 2,y λx x 2,y 2 λx,y y 2 λx 2,y y 2 + λx,y 2 + λx 2,y + λx x 2,y y 2. Therefore, for any fixed x,y >, Hu d khu k,nx,y λx,y N K Hu B 2.7 x,y,σhu 2 as n, where σhu 2 = x 2 x λx,y + y 2 x λx,y y λx,y + 2 y λx,y + 2λx,y 2. λx,y λx,y, 7 x y

9 x λx,y = y λx,y = π/2 π/2 cos φ α dφ cos φ α dφ and2. /2 gx/y /α π/2 cos φ dφ α 2.2 /2 gx/y /α arcsin ρ sinφ + arcsinρ α dφ. Theorem 2.5. Assume that the conditions of Theorem 2. and 2. hold. Further assume 2.2 holds when µ. Suppose k = k n, µ n, k /n n and k FY k /n 2 + AFY n k /n : K, µ, k AF Y k /n n : K, µ =, for K < Then, as n, sup x,y T k k,nx,y λx,y B 2.5 x,yz = op,2.3 where Z N α 2 K B 2.4, α 2 with B 2.4 := B 2.4/s ds, µ, B 2.5/s ds, µ =, B 2.4 s and B 2.5 s are defined in Theorem 2.2 and 2.4 B 2.5 x,y := { π/2 gx/y /α xcos φα lncos φ dφ + gx/y /α ysinφ + arcsin ρ α lnsinφ + arcsinρ dφ arcsin ρ } π/2 π/2 λx,y cosφ α lncosφ dφ cosφ dφ α 2.5. /2 /2 Therefore, for any fixed x,y >, k k,nx,y λx,y d N α 2 K B 2.4 B 2.5 x,y, α 2 B 2.5 x,y 2. 8

10 The next corollary gives the optimal choice of sample fraction for both estimators. As criterion we use the asymptotic mean squared error of k Hu,n and Hu k,n, denoted by amse Huk Hu and amse k, respectively. Corollary 2.6. Assume that the conditions of Theorems 2.4 and 2.5 hold. Further, suppose that AF Y t b t β/α, F Y t 2 + AF Y t b t 2 β/α for some b,b > as t and define b t 2 β/α µ, b 2 t β2/α := b t β/α, µ =. Then amse Hu k Hu = σ 2 Huk Hu + bk Hu /n β/α B 2.7 x,y 2 and amse k = B 2.5 x,y 2 α 2 k + α 2 b 2 k /n β 2/α B Let k opt Hu and kopt denote the optimal sample fraction in the sense of minimizing amse Hu and amse, respectively. Then k opt Hu = k opt = amse opt Hu ασ 2 Hu α/α 2β 2 2βb 2 B2.7 x,y n 2β/α 2β, 2β 2 αb 2 2 α/α 2β2 2 B2.4 n 2β 2 /α 2β 2, α 2β := amse Hu k opt Hu = n 2β/α 2β b 2α/α 2β amse opt σ 2 β/α Hu B2.7 x,y 2α/α 2β 2β/α and := amse k opt = n 2β 2 /α 2β 2 b 2α/α 2β 2 2 α 2β 2 α 2 B 2.5 x,y 2 2αβ2 B 2.4 2α/α 2β2. 9

11 Remark 2.7. Note that k opt is independent of x and y, but k opt Hu depends on x and y. In case of µ =, both amse opt Hu and amseopt depend on n, α, β, ρ, υ, x, y and b, amse opt additionally depends on σ, but the ratio amse opt Hu /amseopt is independent of n and b. Since the optimal k opt is the same as that for Hill s estimator, when µ =, all data-driven methods for choosing the optimal sample fraction for Hill s estimator can be applied to k,nx,y directly. Note that µ is the median of X,Y and the mean of X,Y when α >. Hence, we could estimate µ by the sample median, say µ = µ X, µ Y. Therefore, consider the new estimator k,n x,y with Z i = Xi 2 + Y i 2 replaced by X i µ X 2 + Y i µ Y 2. Similar to the proofs in Ling and Peng 24, we can show that Theorem 2.5 and Corollary 2.6 hold with µ = for this new estimator. 3 Comparisons and Simulation Study First we compare σ 2 Hu,σ2 given in Theorem 2.4 and 2.5. Note that both only depend on α, ρ, x and y. In Figure, we plot the ratio σ 2 α/σ2 Hu α for x = y = as a function of α, and each curve therein corresponds to a different correlation ρ {.,...,.9}. From Figure, we conclude that better in terms of asymptotic variance. is always Second, we compare the two estimators in terms of optimal asymptotic mean squared errors. Since the ratio of the optimal asymptotic mean squared error depends on α, β, Σ, µ, x, y, we consider elliptical distributions with σ = υ =, µ X = µ Y =. In Figure 2, we consider G Fréchetα, i.e. PG > x = exp x α, x >, hence 2. is satisfied with β = α. In Figure 3, we consider G Paretoα, i.e. PG > x = + x α for x >, therefore, 2. is satisfied with β =. Under the above setup, the ratio of optimal asymptotic mean squared errors only depends on α,ρ,x,y. Similar to Figure, we plot the ratio amse opt α/amseopt Hu α for x = y = as a function of α for different ρ s in Figures 2 and 3. We conclude from both Figures that always performs better than Hu in terms of optimal asymptotic mean squared errors as well. Third, we examine the influence of x and y to the ratio of asymptotic mean squared error. We plot the ratio amse opt α/amseopt Hu α for x,y = 2 and G Paretoα in Figure 4, where each curve corresponds to a different pair of α, ρ {2,.9,,.6, 5,.3,,.}. This figure further confirms that Hu always has a smaller optimal asymptotic mean squared error than.

12 and Finally, we study the finite sample behavior of the two estimators Hu x,y x,y. As above, we consider two elliptical distributions with σ = υ =, µ X = µ Y =, G Fréchetα in Figure 5 and G Paretoα in Figure 6. We generate random samples of size n = from these elliptical distributions with α, ρ {2,.9,,.6, 5,.3,,.}, and plot, and, against k =,...,3 for different pairs α, ρ in Figures 5 and Hu 6, where the solid line corresponds to Hu, and the dashed line to,.. This simulation study also confirms the better performance of

13 ρ = σ 2 α/σ2 Hu α α Figure : Ratio σ 2 α/σ2 Hu α for different correlations ρ. amse opt. α/amseopt Hu.5.2 α.25.3 ρ = α Figure 2: Ratio amse opt Hu α/amseopt α for different correlations ρ and β = α

14 amse opt.2.4 α/amseopt Hu.6 α.8. ρ = α Figure 3: Ratio amse opt Hu α/amseopt α for different correlations ρ and β =. amse opt.2 α/amseopt Hu.4.6 α.8 α = 2, ρ =.9 α =, ρ =.6 α = 5, ρ =.3 α =, ρ = arctany/x Figure 4: Ratio amse opt α/amseopt Hu α, x2 + y 2 = 2, for different α, ρ and β =. 3

15 α = 2, ρ =.9 α =, ρ = k Hu Hu λ =.35 λ = k α = 5, ρ =.3 α =, ρ =. Hu Hu λ =.22 λ = k k Figure 5: Mean of estimators Hu, and, for samples of length n = and different k with σ = υ =, µ =, G Fréchetα, and different α, ρ. α = 2, ρ =.9 α =, ρ = Hu Hu k.2 λ =.35 λ = k.5 α = 5, ρ =.3 α =, ρ = Hu Hu λ =.22 λ = k k Figure 6: Mean of estimators Hu, and, for samples of length n = and different k with σ = υ =, µ =, G Paretoα and different α, ρ. 4

16 4 Proofs Proof of Theorem 2.: Without loss of generality, we assume µ =. Let Φ unif,π be independent of G and F i x denote the inverse of F i x, i =, 2. Then, by Hult and Lindskog 22, F X u = σ υ F Y u, for < u <, t F i tx/ F i t = x α, for x > and i =, 2, X,Y d = σg cos Φ,υG sin arcsinρ + Φ. 4.6 Therefore, t P F X X tx, F Y Y ty = t P G cos Φ FY tx/υ,gsinarcsin ρ + Φ FY ty/υ = π/2 P G F Y tx FY ty dφ. 2πt arcsin ρ υ cos φ υ sinarcsinρ + φ 4.7 Note that t = P X > FX t = P G cos Φ > FY t/υ = π/2 P G > F Y t dφ. 2π υ cosφ /2 Further, P G > x/ cos φ/p G x x cosφ α. Hence, in the following formula we can apply the dominated convergence theorem and obtain B 4.8 t := 2πt P G F Y t/υ π/2 t Next, we obtain for φ arcsin ρ, π/2 cos φ dφ α =: /2 B F Y tx υ cos φ FY ty υ sinarcsinρ + φ F Y FY ty sinarcsin ρ + φ. tx cos φ 5

17 Note that sinarcsinρ + φ/ cos φ is strictly increasing, hence its inverse exists and equals arctan ρ/ ρ 2. Therefore, F Y tx FY ty υ cos φ υ sinarcsinρ + φ FY φ arctan ty/f Y tx ρ ρ 2 F =: g Y ty =: hx, y, t. 4.9 tx F Y Since F Y R α, by Proposition.79 of Geluk and de Haan 987 FY tx/fy t t x /α, i.e., It follows from 4.7 and 4.9 that t P F X X tx, F Y Y ty = B 4.8 t π/2 hx,y,t P + hx,y,t P B 4.8 t arcsin ρ hx,y,t t g x/y /α. 4.2 G F Y t FY tx υ cos φ FY t dφ P G FY t/υ FY t G FY ty υ sinarcsinρ + φ FY t dφ. P G FY t/υ 4.2 Hence, the theorem follows from 4.8, 4.2 and Potter s inequality, e.g. see.2 in Geluk and de Haan 987. Proof of Theorem 2.2: Since X,Y d = µ X +σg cos Φ, µ Y +υg sinφ+arcsinρ, we have X 2 + Y 2 d = G 2 d Φ + 2Gd 2 Φ + µ 2 X + µ2 Y. Define d 3 x,φ := d φ d 2 φ + d 2 2φ d φ µ 2 X + µ2 Y x2. 6

18 Since PX 2 + Y 2 t = P G d 3 t, Φ holds for large t, we obtain P X 2 + Y 2 t 2 x 2 x α P X 2 + Y 2 t 2 π P G d 3 tx,φ π = dφ PG t { π [P G d 3 tx,φ = PG t π t d 3tx,φ P G d 3 t,φ dφ x α PG t α ] dφ [ + x α P G d α 3t,φ + x 3t,φ α PG t t d ] dφ π [ α } α + t d 3tx,φ x α t d ] 3t,φ dφ π P G d 3 t,φ dφ PG t π P G d 3 t,φ =: J t + J 2 t + J 3 t dφ. PG t Since ρ <, it is straightforward to check that t t d 3 t,φ = d φ /2, < sup d φ <, φ π Hence, similarly to the proof of Theorem 2., π t and sup d 2 φ < φ π P G d 3 t,φ PG t dφ = π d φ α/2 dφ Similar to the proof of Lemma 5.2 of Draisma et al. 999, for any ε >, there exists t > such that for all t t,d 3 tx,φ t α β P G d 3 tx,φ PG t t d 3tx,φ α 3tx,φ At t d t d 3tx,φ β α α+β { ε + t d 3tx,φ + t d 3tx,φ exp ε 3tx,φ } ln t d

19 Using 4.22, for any fixed x >, we can choose t large enough such that d 3 tx,φ t uniformly for φ [,π]. That is, for any fixed x >, 4.24 holds uniformly for φ [, π]. Therefore, by dominated convergence theorem and 4.23, for x >, J t t At t J 2 t At = π x α β = x α β π x β d φ α β/2 d φ α/2 dφ and4.25 d φ α β/2 d φ α/2 dφ It follows from 4.22 and a Taylor expansion, for x >, that J 3 t = α x tx α + α 2t 2 x α x 2 π π d φ α/2 /2 d 2 φ dφ + o t 2 d φ α/2 [ αd 2 φ 2 + d φµ 2 X + µ 2 Y ] dφ Note that sinφ + arcsinρ = ρ 2 sin φ + ρ cos φ. Then, splitting the integral into [, /2, [/2,, [,π/2, [π/2,π] and using the symmetry of sin and cos, we obtain π d φ α /2 d 2 φ dφ = Hence 2.3 follows from 4.25, 4.26, 4.27 and Note that P X2 + Y 2 > t /PG > t = t π and, since Y d = µ Y + υg sin Φ with Φ unif,π holds, Therefore, we have π PY > t/pg > t = t υα sin φ α dφ. V t F Y t inf inf { y : PG > y t / { y : PG > y t / π d φ α/2 dφ } d φ α/2 dφ υ α π and } sin φ α dφ. 8

20 Hence, t V t F Y t = i.e., since t 2 At t for 2 < β, t = π d φ α/2 /α dφ υ α π sin, φα dφ V t 2 + AV t FY t 2 + AFY t π d φ α/2 2 β /α dφ υ α π sin φα dφ Note that, by Taylor expansion, α V tx = V t x α V tx x +/α V t x/α + o /V t + AV t. 4.3 Therefore, replacing t and x in 2.3 by V t and V tx/v t, respectively, and using 4.29 and 4.3, we obtain 2.4. Let µ X = µ Y =, then J 3 t = and we obtain 2.5. Proof of Theorem 2.3: In order to prove Theorem 2.3, we can assume µ X = µ Y = since λx,y is independent of margins. We also set υ = and give the correction at the end of the proof. Using an upper-triangle decomposition of Σ yields Y d = G sin Φ, where Φ unif,π and is independent of G. Then, write π PY tx PY t P G tx/ sin φ dφ x α = π x α P G t/ sin φ dφ π { P G t/ sin φ π = dφ PG t [ ] } α x α P G t/ sin φ dφ. PG t sin φ Then, by 2., we have for x > PY tx t PY t x α /At π = x αxβ β [ P G tx/ sin φ PG t π sin φ α dφ sin φ α β dφ. 9 ] α x sin φ

21 Replacing t and x in the latter equation by F Y s and F Y sy/f Y s, respectively, we obtain, for y >, F α Y sy y /A FY s = B 2.6 y. 4.3 s s F Y Denote ft := F Y t. Then, by 4.2, we can write t P F X X tx, F Y Y ty π/2 [P G ft ftx cos φ ft = B 4.8 t P G ft hx,y,t α ftx ] dφ ft cos φ π/2 hx,y,t hx,y,t arcsin ρ hx,y,t arcsin ρ [ α ftx xcos φ α] dφ + ft cos φ [P G ft sinarcsin ρ+φ fty ft hx,y, hx,y,t xcos φ α dφ P G ft α fty ] dφ ft sin arcsinρ + φ [ α fty y sin arcsinρ + φ α ] dφ ft sin arcsinρ + φ =: + + hx,y,t hx,y, hx,y, arcsin ρ B 4.8 t y sin arcsin ρ + φ α dφ + y sin arcsinρ + φ α dφ 6 J i t+j 7 +J 8. i= } π/2 hx,y, xcos φ α dφ 4.32 Note that / cos φ and υ is given, using Potter s bound and similar arguments as in the proof of Lemma 5.2 of Draisma et al. 999, for any ε >, there exists some small t > such that for all ft ft, ftx ft and 2

22 φ [/2,π/2] α P G ftx /P G ft ftx cos φ ftcos φ Aft β α ftx ftx ftcos φ ft cos φ β α α+β { ftx ftx ε + + exp ε ft cosφ ft cos φ ln ftx }, ft cos φ 4.33 and for all t t and tx t, εx /α exp ε log x ftx ft + ǫx /α expε log x Since ft t and t t imply that ftx t and tx t for all x, respectively, by 4.33, 4.34, 4.2 and dominated convergence, we have t J t Aft = x β π/2 hx,y, [ x β/α cosφ α β cosφ α] dφ 4.35 holds for all x,y and uniformly on S + 2. Similarly, t J 4 t Aft = y β hx,y, arcsin ρ [ y β/α sinφ + arcsinρ α β 4.36 sinφ+arcsinρ α] dφ holds for all x,y and uniformly on S + 2. Using 4.3 and a way similar to the proof of Lemma 5.2 of Draisma et al. 999, for any ε >, there exists t > such that for all t t and tx t FY tx/f Y t α x B AFY 2.6 x s ε C +C 2 x+c 3 x β/α expε ln x,

23 where the constants C >,C 2 >,C 3 > are independent of x and t. Hence, it follows from 4.2 and 4.37 that t t J 2 t π/2 A FY t = B 2.6 x cos φ α dφ and 4.38 hx,y, J 5 t A F Y t = B 2.6 y holds for all x,y and uniformly on S + 2. hx,y, arcsin ρ sinφ + arcsinρ α dφ 4.39 Note that fty α y Aft ftx x [ fty α = y Aft x ft x fty ftx α ftx α x] ft t x B 2.6y y x 2 B 2.6x. Similar to the proof of Lemma 5.2 of Draisma et al. 999, for any ε >, there exists t > such that for all t t,tx t,ty t fty α y Aft ftx x x B 2.6y + y x B 2.6x 2 x ε C + C 2 y + C 3 y β/α ε log e y + x + x y x C + C 2 x + C 3 x β/α expε log x y x expε logy/x C + C 2 x + C 3 x β/α expε log x,4.4 where constants C >,C 2 >,C 3 > are independent of t,x,y. Using 4.4, sup z g z /α z 2/α < sup z g z /α < sup z [singz /α + arcsinρ] α z < 22

24 and applying a Taylor expansion to gz /α, we can show that and t J 3 t Aft = t = x α gx/y /α xcos φ α dφ Aft gfty/ftx [ cos g x/y /α] α g t J 6 t Aft = t Aft x/y /α B 2.6 y gfty/ftx gx/y /α y B 2.6x x x y ysinφ + arcsin ρ α dφ = y [ sin g x/y /α ] α + arcsinρ g x/y /α α /α 4.4 B2.6 y y B 2.6x x x y /α 4.42 holds for all x,y and uniformly on S 2 +. Since [ x cos g x/y /α] α [ = y sin g x/y /α α + arcsin ρ],4.43 we obtain t J 3 t + J 6 t/aft =. By Theorem 2., λx,y = J 7 + J 8 /B 4.8, hence t Aft = t Aft = λx,y β B 4.8 t J 7 + J 8 λx,y λx,y B4.8 t B 4.8 B 4.8 t π/2 cosφ α cos φ β dφ /2 which obviously holds uniformly on S + 2 since sup S + 2 π/2 cosφ dφ α, /2 λx,y <. Note that 4.44 A F Y t/a F υy t t υ β

25 Hence the theorem follows from 4.35, 4.36, 4.38, 4.39, 4.4, 4.42, 4.44 and Proof of Theorem 2.4: Similar to Huang 992 or Einmahl, de Haan and Li 24, we have, as n, where sup x,y T = o p, k Hu { x+y Hu k Hu,nx,y lx,y Bx,y = Wx,y } K Hu B 2.7 x,y Bx,y λx,y Wx, λx,y W,y, x y and Wx,y is a Wiener process with zero mean and covariance structure E Wx,y Wx 2,y 2 = lx x 2,y lx x 2,y 2 lx,y y 2 + lx 2,y y 2 lx,y 2 lx 2,y lx x 2,y y 2. Hence 2.9 follows from λx,y = x + y lx,y. It is straightforward to check that 2., 2. and 2.2 hold. Note that the result can also be obtained from Schmidt and Stadtmüller 25 by taking the bias into account. Proof of Theorem 2.5: The result follows directly from k α H k,n α see de Haan and Peng 998, τ n τ = o p the expression of λx,y given in Theorem 2.. d N α 2 K B 2.4,α 2 k /2 and the delta method for Acknowledgments Peng s research was supported by NSF grant DMS and a Humboldt Research Fellowship. 24

26 References [] Draisma, G., de Haan, L., Peng, L. and Pereira, T. 999 A Bootstrap-based Method to Achieve Optimality in Estimating the Extremevalue Index. Extremes 24, pp [2] Drees, H. and Huang, X. 998 Best Attainable Rates of Convergence for Estimates of the Stable Tail Dependence Functions. J. Mult. Anal. 64, pp [3] Einmahl, J., de Haan, L. and Huang, X. 993 Estimating a Multidimensional Extreme Value Distribution. J. Mult. Anal. 47, pp [4] Einmahl, J., de Haan, L. and Li, D. 24 Weighted Approximations of Tail Copula Processes with Application to Testing the Multivariate Extreme Condition. Submitted. Available at [5] Einmahl, J., de Haan, L. and Piterbarg, V. 2 Nonparametric Estimation of the Spectral Measure of an Extreme Value Distribution. Ann. Statist. 295, pp [6] Einmahl, J., de Haan, L. and Sinha, A. 997 Estimating the Spectral Measure of an Extreme Value Distribution. Stochastic Process. Appl. 72, pp [7] Geluk, J. and de Haan, L. 987 Regular Variation, Extensions and Tauberian Theorems. CWI Tract 4. [8] de Haan, L. and Peng, L. 998 Comparison of Tail Index Estimators. Statist. Neerlandica 52, pp [9] de Haan, L. and Resnick, S. 993 Estimating the Limit Distribution of Multivariate Extremes. Commun. Statist. Stochastic Models 92, pp [] de Haan, L. and Stadtmüller, U. 996 Generalized Regular Variation of Second Order. J. Australian Math. Soc. Ser. A 63, pp [] Hill, B. 975 A Simple General Approach to Inference About the Tail of a Distribution. Ann. Statist. 35, pp

27 [2] Huang, X. 992 Statistics of Bivariate Extremes. Ph.D. Thesis, Erasmus University Rotterdam, Tinbergen Research Institute, no. 22. [3] Hult, H. and Lindskog, F. 22 Multivariate Extremes, Aggregation and Dependence in liptical Distributions. Adv. Appl. Prob. 343, pp [4] Joe, H. 997 Multivariate Models and Dependence Concepts. Chapman&Hall, London. [5] Ledford, A. and Tawn, J. 997 Modelling Dependence within Joint Tail Regions. Statistics for near Independence in Multivariate Extreme Values. J. Royal Statist. Soc. B, 592, pp [6] Ling, S. and Peng, L. 24 Hill s Estimator for the Tail Index of an ARMA Model. Journal of Statistical Planning and Inference, 232, pp [7] Resnick, S. 987 Extreme Values, Regular Variations and Point Processes. Springer, New York. [8] Schmidt, R. 23 Credit Risk Modelling and Estimation via liptical Copulae. In Credit Risk: Measurement, Evaluation and Management, ed. Bohl, Nakhaeizadeh, Rachev, Ridder and Vollmer. Physica Verlag Heidelberg, pp [9] Schmidt, R. and Stadtmüller, U. 25 Nonparametric Estimation of Tail Dependence. Scand. J. of Stat. To appear. [2] Tawn, J. 988 Bivariate Extreme Value Theory: Models and Estimation. Biometrika 753,