INFERENCE FOR ASYMPTOTICALLY INDEPENDENT SAMPLES OF EXTREMES

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Inference for asymptotically independent samples of extremes

Armelle Guillou, Simone A. Padoan, Stefano Rizzelli

To cite this version:

Armelle Guillou, Simone A. Padoan, Stefano Rizzelli. Inference for asymptotically indepen- dent samples of extremes. Journal of Multivariate Analysis, Elsevier, 2018, 167, pp.114-135.

�10.1016/j.jmva.2018.04.009�. �hal-01553839v3�

Inference for asymptotically independent samples of extremes

Armelle Guillou ⁽¹⁾ , Simone A. Padoan ⁽²⁾ and Stefano Rizzelli ⁽²⁾ .

(1)

Institut Recherche Math´ematique Avanc´ee, UMR 7501 Universit´e de Strasbourg et CNRS

7 rue Ren´e Descartes 67084 Strasbourg cedex

France

e-mail: armelle.guillou@math.unistra.fr

(2)

Department of Decision Sciences, Bocconi University of Milan

via Roentgen 1 20136 Milano, Italy

e-mail: simone.padoan@unibocconi.it stefano.rizzelli@phd.unibocconi.it

Abstract: An important topic of the multivariate extreme-value theory is to develop probabilistic models and sta- tistical methods to describe and measure the strength of dependence among extreme observations. The theory is well established for data whose dependence structure is compatible with that of asymptotically dependent models.

On the contrary, in many applications data do not comply with asymptotically dependent models and thus new tools are required. This article contributes to the methodological development of such a context, by considering a componentwise maxima approach. First we propose a statistical test based on the classical Pickands dependence function to verify whether asymptotic dependence or independence holds. Then, we present a new Pickands de- pendence function to describe the extremal dependence under asymptotic independence. Finally, we propose an estimator of the latter, we establish its main asymptotic properties and we illustrate its performance by a simulation study.

MSC 2010 subject classifications: Primary 62G32, 62G05, 62G20; secondary 60F05, 60G70.

Keywords and phrases: Extremal dependence, extreme-value copula, nonparametric estimation, Pickands depen- dence function.

1. Introduction

Multivariate extreme-value theory provides the mathematical foundation for performing real data analysis of rare events. To characterize the joint upper tail of a multivariate distribution, two di ff erent approaches can be used: either by considering the componentwise maxima, or all the observations above a high threshold. A description of these methodologies can be found for instance in Coles (2001, Ch. 8), Beirlant et al. (2004, Ch. 8-9), de Haan and Ferreira (2006, Ch. 6) and Resnick (2007, Ch. 6), among others. Unfortunately the flexibility of the dependence structures provided by the classical theory of multivariate extreme-values may not be su ffi cient for statistical modelling (see e.g. Ledford and Tawn, 1996, 1997). To solve this issue, di ff erent coe ffi cients of tail dependence or probabilistic models have been introduced. They allow to govern / describe the strength of the extremal dependence. In this paper, we are particularly interested in the notion of asymptotic independence which is common in real data analysis. This concept can be defined as follows.

Let Y be a multivariate random vector of dimension d, with distribution function F and marginals F j , 1 ≤ j ≤ d.

We say that F is in the max-domain of attraction of a multivariate extreme-value distribution G, if there exist sequences of constants a _n > 0 and b _n ∈ R ^d such that

n→∞ lim F ⁿ (a _n y + b _n ) = G(y),

for all y ∈ R ^d . Under this condition, a particular case arises when G is equal to the product of its marginal distributions. In this setting, we say that Y satisfies the property of asymptotic independence (or tail independence)

∗

The first author is supported by a research grant (VKR023480) from VILLUM FONDEN

which is equivalent to say that the elements of Y are asymptotically independent in the upper tail, i.e.

u→1 lim Pr(F j (Y j ) > u | F i (Y i ) > u) = 0

for all 1 ≤ i , j ≤ d. On the contrary, if the above limit is positive, then the elements of Y are called asymptotically dependent. The classical theory expects asymptotic dependence and independence as the only two possible scenar- ios, conceiving extremes as independent in the second case. Many e ff orts have been made to characterize a residual tail dependence in the data (if there is any) by o ff ering new dependence coe ffi cients or probabilistic and statisti- cal models under asymptotic independence, see Ledford and Tawn (1996), Coles (2001, Ch 8.4), Resnick (2002), Maulik and Resnick (2004), Ramos and Ledford (2009, 2011), Wadsworth and Tawn (2013) and Wadsworth et al. (2017), to name a few. If we restrict our framework to the dimension d = 2, several statistical tests for check- ing asymptotic independence or tail independence have been proposed, among them, Ledford and Tawn (1996), Draisma et al. (2004), H¨usler and Li (2009) and Falk et al. (2010, Ch. 6.5) and the references therein. However, the extension to dimensions higher than 2 are still in its infancy. Recent proposals are based on the kth largest order statistics of the sample. Although these approaches are simple to implement, the performance of the resulting tests depends strongly on the choice of k, see e.g. Kiriliouk et al. (2016).

In this paper, we propose in Section 2 an alternative approach to test asymptotic independence for an arbitrary dimension d ≥ 2, based on the componentwise maxima. We illustrate the performance of our proposal up to di- mension d = 4. Then, using again the componentwise maxima approach and in particular the framework proposed by Ramos and Ledford (2011), we introduce, in Section 3, a new dependence function similar to the well-known Pickands dependence function which allows us to measure the residual dependence under asymptotic indepen- dence. Finally, we estimate this new dependence function and we establish the main asymptotic properties of the estimator. By means of a simulation study, its good performance is highlighted. A discussion on the methodological assumptions and possible extensions of our work ends the paper. All the proofs are postponed to the appendix.

Throughout the paper, the following notations are used. For any arbitrary dimension d and f : X ⊂ R ^d → R, we set k f k _∞ = sup _x∈X f (x). We denote by ` <sup>∞</sup> (X) the space of all bounded real-valued functions on X. The symbol “ ” stands for convergence in distribution of random vectors, but also for weak convergence of bounded real-valued functions in ` ^∞ (X), the di ff erence will be clear from the context.

2. A test for asymptotic independence

A d-dimensional random vector X = (X ₁ , . . . , X d ) follows the law of a multivariate extreme-value distribution if the one-dimensional marginal distributions, G _j (x) = Pr(X _j ≤ x) for all x ∈ R, j = 1, . . . , d, are Generalized Extreme-Value (GEV) distributions, and the joint distribution takes the form

G(x) = C G 1 (x 1 ), . . . , G _d (x _d ) , x ∈ R ^d , where C is an extreme-value copula, i.e.,

C(u) = exp

− V

(− log u 1 ) ⁻¹ , . . . , (− log u d ) ⁻¹

, u ∈ (0, 1] ^d ,

with V : (0, ∞] ^d → [0, ∞) (see de Haan and Ferreira, 2006, Ch. 1, 6, for details). Consider the map L : [0, ∞) ^d 7→

[0, ∞), defined by L(z) : = V (1/z) with z = 1/y for y ∈ (0, ∞] ^d . The function L is known as the stable tail dependence function. As it is a homogeneous function of order one, i.e. L(a z) = aL(z) for all a > 0, we have

L(z) = (z 1 + · · · + z d ) A(t), z ∈ [0, ∞) ^d ,

with t j = z j /(z 1 + · · · + z d ) for j = 2, . . . , d, t 1 = 1 − t 2 − · · · − t d , and A is the restriction of L into the d-dimensional unit simplex,

S _d : = n

(v ₁ , . . . , v d ) ∈ [0, 1] ^d : v 1 + · · · + v d = 1 o

.

A(t) ≤ 1 for all t ∈ S _d , with lower and upper bounds corresponding to the complete dependence and independence cases, respectively (see Falk et al., 2010, Ch. 4, for details). Thus, estimating this Pickands dependence function is crucial for analysing multivariate extremes, and it has been an extensively discussed topic in the literature, see Kl¨uppelberg and May (2006), Zhang et al. (2008), Gudendorf and Segers (2011), Cap´era`a et al. (1997), Berghaus et al. (2013) or Vettori et al. (2017), among others.

2.1. A slightly modified version of the Pickands dependence estimator proposed by Marcon et al. (2017) This estimator is based on the madogram concept, a notion borrowed from geostatistics in order to capture the spatial structure. Starting from independent and identically distributed (i.i.d.) copies X 1 , ..., X n , of X, our estimator is defined as

b A n (t) : = b ν n (t) + c(t)

1 − b ν n (t) − c(t) (2.1)

where

b ν n (t) : = 1 n

n

X

i = 1



 

 

 



d

_

j = 1

n G ⁽¹⁾ _n,j (X i,j ) o 1/t

_j

− 1 d

d

X

j = 1

n G ⁽¹⁾ _n,j (X i, j ) o 1/t

_j



 

 

 



(2.2)

c(t) : = 1 d

d

X

j = 1

t _j 1 + t j

with

G ^(a) _n,j (X i, j ) : = G n,j (X i,j )



 

 

  1 + a

a 1 n

n

X

k = 1

G ^1/a _n,j (X k,j )



 

 

 

−a

, j = 1, . . . , d, for a > 0, and the empirical distribution functions denoted by

G _n,j (x) : = 1 n

n

X

i = 1

1l {X

_i,j

≤x} , j = 1, . . . , d.

By convention, here u ^1/0 = 0 for 0 < u < 1. The estimator (2.2) is a slightly modified version of that proposed in Marcon et al. (2017), with G ⁽¹⁾ _n,j in place of G _n,j which ensures that the new Pickands estimator A b _n now satisfies b A n (e j ) = 1 for all j = 1, . . . , d, where e j = (0, . . . , 0, 1, 0, . . . , 0) is the jth canonical unit vector (see Appendix A.1).

This is a necessary condition that a function needs to satisfy in order to be a valid Pickands dependence function (see e.g. Marcon et al., 2017). Although, as established in Appendix A.1, our modified estimator shares the same asymptotic properties as the estimator discussed in Marcon et al. (2017), our modification greatly improves the latter for finite samples.

2.2. Construction of our statistical test

Using our estimator for A, we want now to construct a statistical test to check asymptotic independence in dimen- sions higher than or equal to two. To this aim, we consider the following system of hypotheses

( H ₀ : A(t) = 1, ∀ t ∈ S _d H ₁ : A(t) < 1, for some t ∈ S _d .

Note that H 0 means that all the components of X are asymptotically independent, whereas under H 1 some elements of X are asymptotically dependent.

Assuming that the extreme-value copula C has continuous partial derivatives over the sets {u ∈ [0, 1] ^d : 0 < u i < 1}, by Theorem 2.4 in Marcon et al. (2017) and according to Appendix A.1, under H 0 we have that

√ n

A b _n (t) − 1

t∈S

d

−4 Z 1

0 A (v ^t

¹

, . . . , v ^t

^d

)dv

!

t∈S

, as n → ∞, (2.3)

where A is a centered Gaussian process on [0, 1] ^d with continuous sample paths and covariance function equal to Cov( A (u), A (v)) =

d

Y

j = 1

u _j ∧ v _j −

d

X

j = 1



 

 

 



u _j ∧ v _j Y

i , j

u _i v _i



 

 

 

 + (d − 1)

d

Y

j = 1

u _j v _j .

As a consequence, by the continuous mapping theorem (see e.g. van der Vaart, 2000, Ch. 2.1) it follows that b S n := sup

t∈S

d

√ n

A b n (t) − 1

S := sup

t∈S

d

4

Z 1 0

A (v ^t

¹

, . . . , v ^t

^d

) dv

, as n → ∞ .

This convergence result can be used as the cornerstone of our test. Denoting by Q S (α), α ∈ (0, 1), the (1 − α)-quantile function for the distribution of the random variable S , H 0 can be rejected with approximately α%

significance level whenever b s n , the observed value of b S n , exceeds Q S (α). Unfortunately, there is no closed form for the function Q _S (α), however an approximation can still be computed with a Monte Carlo simulation as follows.

Note that for any u, v ∈ [0, 1] and t, w ∈ S _d , the covariance function of the Gaussian process A in (2.3), evaluated at the indexes u ^t , v ^w ∈ [0, 1] ^d , is equal to

Cov( A (v ^t

¹

, . . . , v ^t

^d

), A (u ^w

¹

, . . . , u ^w

^d

)) =

d

Y

j = 1

(v ^t

^j

∧ u ^w

^j

) −

d

X

j = 1

(v ^t

^j

∧ u ^w

^j

)v ^1−t

^j

u ^1−w

^j

+ (d − 1)uv. (2.4) Thus, for any fixed α ∈ (0, 1), an approximation of the quantile Q S (α) can be obtained by adhering to the following four steps:

1. Divide the unit interval (0, 1) and the simplex S _d in p and m equally spaced points, where p and m are positive integers. Let v 1 , . . . , v m and t 1 , . . . , t p be the two sequences of points partitioning (0, 1) and S _d , respectively.

The sequences v 1 , . . . , v m and t ₁ , . . . , t _p form a finite sequence of positions v ^t r

^k,1

, . . . , v ^t r

^k,d

∈ [0, 1] ^d , with r = 1, . . . , m and k = 1, . . . , p, on which the process A is simulated.

2. Sample n ^∗ realizations

x _i (v ^t ₁

^1,1

, . . . , v ^t ₁

^1,d

), . . . , x _i (v ^t _m

^p,1

, . . . , v ^t _m

^p,d

), i = 1, . . . , n ^∗ ,

of a zero-mean Gaussian process at v ^t _r

^k,1

, . . . , v ^t _r

^k,d

, for r = 1, . . . , m and k = 1, . . . , p, with a (mp × mp) variance-covariance matrix defined through the covariance function in (2.4).

3. Simulate samples that approximately follow the distribution of the random variable S , the integral and the sup in S being approximated by a sum and the max for su ffi ciently large values of m and p. This leads to the realizations

e s i = max

1≤k≤p 4 1 m

m

X

r = 1

x i (v ^t _r

^k

^,1 , . . . , v ^t _r

^k

^,d )

, i = 1, . . . , n ^∗ .

4. An approximation of the quantile Q S (α), denoted by Q e S (α), can then be obtained by computing the sample quantile of the realizations e s 1 , . . . , e s n

^∗

for su ffi ciently large n ^∗ .

2.3. Numerical results

We illustrate the performance of our statistical test through a simulation study. Precisely, we estimate some values of the significance level α and the power 1 − β of the test by computing the empirical proportion of simulated samples under the null hypothesis and the alternative hypothesis that rejected the null hypothesis, respectively. For simplicity we focus on the significance levels α = 0.05 and 0.01. The study consists of five experiments.

First experiment: In order to perform the first experiment, in a first step we compute the approximated quantile

Q e S (α), for a given α, following our algorithm. The quality of approximation relies on the values of the indexes

m, p and n ^∗ . Clearly, the larger their values are, the more accurate the approximation is. We set n ^∗ = 500 000.

Table 1

Estimated significance levels α. From left to right: the dimension of X, the true significance level, the approximate asymptotic (1 −α)-quantile, the empirical proportion of simulated samples under H

₀

that rejected the null hypothesis and the empirical (1 − α)-quantile. Here ψ = 1.

α ˆ Q

bSn

(α)

n n

d α Q e

S

(α) 25 50 100 200 25 50 100 200

2 0.05 0.960 0.0380 0.0460 0.0512 0.0524 0.9190 0.9393 0.9512 0.9541 0.01 1.204 0.0060 0.0082 0.0102 0.0102 1.1359 1.1739 1.1926 1.1992 3 0.05 1.300 0.0364 0.0452 0.0508 0.0574 1.2540 1.2755 1.3036 1.3210 0.01 1.540 0.0056 0.0068 0.0084 0.0092 1.4126 1.4904 1.5295 1.5601 4 0.05 1.480 0.0398 0.0454 0.0548 0.0576 1.5312 1.5508 1.5883 1.5745 0.01 1.740 0.0064 0.0082 0.0096 0.0126 1.7715 1.7135 1.7849 1.7867

better value for these indexes when the value of Q e S (α) does not increase anymore, up to the second decimal. The calculation of Q e _S requires a considerable computational e ff ort, therefore we derive its values only for a dimension d = 2, 3, 4 of the vector X.

In a second step, we compute the rejection rates. We focus on the multivariate logistic extreme-value model intro- duced by Tawn (1990), with dependence parameter ψ ∈ (0, 1], ψ = 1 corresponding to independent components of X, whereas complete dependence can be reached when ψ → 0. We consider 20 equally spaced values of ψ in (0, 1]. For each of them, we simulate n independent observations from a logistic extreme-value distribution with unit Fr´echet margins. Then we estimate the Pickands dependence function by (2.1) and we compute b s n . We repeat this task 5000 times and we compute the proportion of times that b s n > Q e S (α). This experiment is repeated for dif- ferent values of the samples size n and di ff erent dimension d of X. The middle part of Table 1 reports the estimated values of the significance levels α in the case where ψ = 1.

We see that accurate estimates of α are already obtained with the sample size n = 50, indicating a good performance of our statistical test. Figure 1 displays the estimated powers of the test. In the first and second rows the results obtained with α = 0.05 and α = 0.01 are reported, respectively. The panels from left to right illustrate the results for the dimensions 2, 3 and 4. Once again, the test shows a good performance already with the sample size n = 50.

Indeed in the case d = 2 we see that the power of the test reaches 1 with mild dependence levels, i.e. ψ = 0.5. This figure also outlines that the power of the test improves as the dimension of X increases and that, as expected, for any fixed dimension d = 2, 3, 4, it also improves as the sample size increases.

Second experiment: We repeat the second step of the first experiment approximating the test statistic’s distribu- tion, under the null hypothesis, via a Monte Carlo approach. Precisely, we simulate n values from d independent univariate Fr´echet distributions, then we estimate the Pickands dependence function by (2.1) and we compute b s n . We repeat this task 5000 times and we compute the empirical quantile, for a given α, denoted by Q

b S

n

(α). The right-hand side of Table 1 reports the values of them for di ff erent values of n and d. We see that the empirical quantiles rapidly approach the asymptotic quantiles, as the sample size increases. Already with the sample size n = 50 the two types of quantiles are very close. The third and fourth lines of Figure 1 display the comparison between the estimates of 1 − β obtained with the two types of quantiles, but also the estimates of α since ψ = 1 corresponds to independent components and thus the proportion of rejection reported by the figures represents an approximation of α in that case. Since the conclusions are the same for both values α = 0.01 and α = 0.05, only the latter are reported. It follows that the inferential results obtained with the empirical quantiles are very close to those obtained with the asymptotic quantiles, already with the sample size n = 50.

Third experiment: We repeat the second experiment using the Genest-R´emillard (GR) rank-based statistical test (Genest and R´emillard, 2004) and our proposed test with the Cap´era`a-Foug`eres-Genest (CFG) estimator (Cap´era`a et al., 1997; Zhang et al., 2008) of the Pickands dependence function in place of (2.1). Figure 2 shows the estimated powers obtained with the GR test and our test (with both the CFG and the Madogram-based (2.1) estimators). For brevity, we show the results for α = 0.05 and the sample sizes n = 25 and n = 50. We see that our test always outperforms the GR test, with the best results provided with the CFG estimator in the case of a dimension equal to 2, whereas, in higher dimensions, similar results can be reached with either the Madogram-based estimator or the CFG.

Fourth experiment: We repeat the third experiment by sampling from three alternative distributions. In the first

case, we draw samples from a three-dimensional random vector with a pair that follows the logistic extreme-value distribution and where the last variable is independent from the other two. In the second case, we consider a four-dimensional random vector with two pairs that follow the logistic extreme-value distribution and where the components of one pair are independent from each component of the second pair. In the last case, we consider a four-dimensional random vector, where one pair with independent components and each of these independent from the components of the other pair that follows the logistic extreme-value distribution. The results are collected in the third (n = 25) and fourth row (n = 50) of Figure 2. In these cases we see that our test loses power and provides inferential results very similar to those provided by the GR test, however the latter outperforms our test in the case of the largest number of independent variables.

Fifth experiment: We consider the multivariate inverted symmetric logistic model (see e.g. Ledford and Tawn, 1997; Wadsworth et al., 2017), with dependence parameter ψ ∈ (0, 1], ψ = 1 corresponding to exact independence of the components of X, whereas asymptotic dependence is reached as ψ → 0. This time, we consider 10 equally spaced values of ψ in (0, 1]. For each of them, we simulate 366 values (for similarity with annual maxima) from an inverted logistic distribution with exponential margins. Then, we compute the normalized componentwise maxima and we repeat this procedure in order to obtain n normalized maxima from which we estimate the Pickands de- pendence function using (2.1) and we calculate b s n . We repeat this task 5000 times and we compute the proportion of times that b s n > Q S

_n

(0.05). This procedure has been done for di ff erent values of d and n and the results are summarized in Table 2.

Table 2

Estimated significance levels α. From left to right: the dimension of X, the sample size and the empirical proportion of simulated samples under H

₀

that rejected the null hypothesis for di ff erent values of ψ.

ψ

d n 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

2 25 0.0522 0.0468 0.0578 0.0580 0.0584 0.0842 0.1294 0.2730 0.6372 0.9920 50 0.0554 0.0524 0.0542 0.0562 0.0590 0.0980 0.1772 0.4538 0.8932 1.0000 100 0.0476 0.0470 0.0506 0.0561 0.0594 0.1208 0.2938 0.7128 0.9962 1.0000 200 0.0486 0.0570 0.0568 0.0560 0.0601 0.1856 0.4934 0.9482 1.0000 1.0000 3 25 0.0470 0.0508 0.0522 0.0582 0.0808 0.1076 0.2190 0.4234 0.8604 0.9990 50 0.0542 0.0538 0.0528 0.0589 0.8320 0.1604 0.3606 0.7424 0.9950 1.0000 100 0.0536 0.0468 0.0550 0.0594 0.0922 0.2084 0.5274 0.9486 1.0000 1.0000 200 0.0540 0.0500 0.0506 0.0652 0.1242 0.3050 0.8096 0.9996 1.0000 1.0000 4 25 0.0488 0.0418 0.0448 0.0504 0.0692 0.1336 0.2676 0.6332 0.9582 1.0000 50 0.0452 0.0438 0.0536 0.0574 0.1018 0.1854 0.4736 0.8852 0.9996 1.0000 100 0.0496 0.0484 0.0468 0.0598 0.1130 0.2746 0.7052 0.9920 1.0000 1.0000 200 0.0486 0.0448 0.0560 0.0770 0.1596 0.4768 0.9440 1.0000 1.0000 1.0000

With d = 2, the rejection rates are close to 0.05 whenever ψ is larger than 0.5. Otherwise, the rejection rate is greater than 0.05 and it reaches 1 when ψ approaches 0. In these cases, it can be observed that the normalized maxima show quite a strong dependence, which indeed seems that of an asymptotically dependent model rather than an asymptotically independent one. The strength of the dependence is reduced whenever the normalized maxima are computed on sequences larger than 366, resulting in improvements in the performances of our test.

The test performance deteriorates as the dimension of X increases.

In conclusion this study highlights the good performance of our statistical test in detecting the exact indepen-

dence of sample maxima. However, our test is also useful to detect if multivariate data are asymptotically indepen-

dent as long as there is a weak dependence within the case of asymptotic independence. On the contrary, in the

cases of strong dependence within asymptotic independence our test fails in detecting the data as asymptotically

independent. Clearly, these are the most naturally di ffi cult cases to detect and more specific tools are required.

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0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

d=2

ψ

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Fig 1. Estimated power functions. Points report the empirical proportion of simulated samples under H

1

that rejected H

₀

as a function of ψ.

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Fig 2. Estimated power functions. Points are the empirical proportion of simulated samples under H

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3. Asymptotic independence for componentwise maxima

Being able to test asymptotic independence versus asymptotic dependence is obviously important. However, since asymptotic independence often arises in applications, it is also crucial to develop some general models that accom- modate both situations. In this section, we consider the framework of Ramos and Ledford (2009) (see also Ledford and Tawn, 1997). More precisely, if Y is a d-dimensional random vector with common unit Fr´echet margins, i.e.

Pr(Y _j ≤ y) = e ^−1/y for every y > 0 and 1 ≤ j ≤ d, this theory relies on the joint survival function of Y which is as- sumed to be multivariate regularly varying with index −1/η, where η ∈ (0, 1], i.e. Pr(Y > y) = τ(y)(y ₁ · · · · · y _d ) ^−1/dη with τ a slowly varying function satisfying

r→∞ lim

τ(r y 1 , . . . , r y d ) τ(r, . . . , r) = g(y)

for all y ∈ (0, ∞] ^d . The function g here is homogeneous of order 0, i.e. such that g(a y 1 , . . . , a y d ) = g(y 1 , . . . , y d ) for any a > 0. This framework implies that the conditional joint survival function admits the following limit representation for every y ≥ 1, the vector of ones,

u→∞ lim Pr(Y > uy|Y > u1) = lim

u→∞

Pr(Y > uy) Pr(Y > u1) = η Z

S

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d

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w j

y j

! 1/η

dH _η (w), (3.1)

where H η is a non-negative measure satisfying the condition η Z

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d

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w ^1/η _j dH _η (w) = 1. (3.2)

This measure H η is a particular case of the hidden angular measure introduced by Resnick (2002) (see also Maulik and Resnick, 2004) when η < 1 and it is a rescaled version of the classical angular measure when η = 1, see Ramos and Ledford (2009) for details. According to Ramos and Ledford (2011) we assume that H η is a finite measure. We recall that η is the so-called coe ffi cient of tail dependence, which measures the level of dependence within the asymptotic independence framework. Specifically, η = 1 corresponds to the case of asymptotic depen- dence, whereas η < 1 corresponds to the case of asymptotic independence. More precisely, when the coe ffi cient η falls in the following sets: (1/d, 1), {1/d} or (0, 1/d), then we say that among the variables there is a positive association, independence or negative association, respectively, within asymptotic independence (see e.g. Ledford and Tawn, 1996).

3.1. A η−Pickands dependence function

Consider now, n i.i.d. copies Y 1 , . . . , Y n of Y and for a small ε > 0, define M n,ε = (M n,1,ε , . . . , M n,d,ε ) as the vector of componentwise maxima, precisely

M n, j,ε = _

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n

(ε)

Y i, j , j = 1, . . . , d,

with I n (ε) : = {1 ≤ i ≤ n : Y i > 1ε}. Let b n be a sequence of normalizing constants defined by the equation nPr(Y > b n ) = 1. Then, di ff erently from the classical theory (e.g. de Haan and Ferreira, 2006, Ch. 6), here the limiting distribution for the normalized vector of componentwise maxima M n,ε is obtained as

G η (y) := lim

ε→0 lim

n→∞ Pr(M _n,b

_n

_ε ≤ b n y), y ∈ (0, ∞ ] ^d , (3.3)

see Ramos and Ledford (2011) for details. When a limiting distribution exists with nondegenerate margins, then

G _η is called a multivariate η-extreme-value distribution. Specifically, a d-dimensional random vector Z follows