On the tail dependence in bivariate hydrological frequency analysis

Abstract In Bivariate Frequency Analysis (BFA) of hydrological events, the study and quantification of the dependence between several variables of interest is commonly carried out through Pearson’s correlation (r), Kendall’s tau (τ) or Spearman’s rho (ρ). These measures provide an overall evaluation of the dependence. However, in BFA, the focus is on the extreme events which occur on the tail of the distribution. Therefore, these measures are not appropriate to quantify the dependence in the tail distribution. To quantify such a risk, in Extreme Value Analysis (EVA), a number of concepts and methods are available but are not appropriately employed in hydrological BFA. In the present paper, we study the tail dependence measures with their nonparametric estimations. In order to cover a wide range of possible cases, an application dealing with bivariate flood characteristics (peak flow, flood volume and event duration) is carried out on three gauging sites in Canada. Results show that r, τ and ρ are inadequate to quantify the extreme risk and to reflect the dependence characteristics in the tail. In addition, the upper tail dependence measure, commonly employed in hydrology, is shown not to be always appropriate especially when considered alone: it can lead to an overestimation or underestimation of the risk. Therefore, for an effective risk assessment, it is recommended to consider more than one tail dependence measure.


Introduction
Given economic, social and scienti c issues related to oods, storms and droughts, no serious debate on these notions can be conducted without a re ection on the extreme nature of these events [e.g. 13,47]. They require an accurate modelling and an appropriate analysis. In order to evaluate hydrological risk, some studies advocate univariate analysis based mainly on ood peaks [e.g. 6]. Nevertheless, hydrological processes are characterized by several variables. For instance, oods are mainly described with three variables (peak ow, ood volume and event duration) obtained from the hydrograph [e.g. 69,74]. Thus, an e ective risk assessment cannot be conducted by studying each variable separately since this does not take into account the dependence between variables and can lead to an overestimation or underestimation of the risk [e.g. 11,18,52,59]. In such a situation, copulae are widely employed [e.g. 9,28]. In hydrology, the quanti cation of the degree of dependence between the underlying variables with an indicator value in a scalar format is fundamental [63].
During the last years, the study of the dependence of hydrometeorological variables has gained increasing attention in hydrological risk assessment [see 11, and references therein]. In this framework, common measures such as Pearson's correlation r, Kendall's τ and Spearman's ρ have been largely employed by hydrologists. However, these indicators are not always appropriate for a proper understanding of dependencies in Bivariate Frequency Analysis (BFA) of extreme events [see e.g. 22, for a study in nancial markets] since they cover the whole distribution without focusing on the tail of the distribution where extreme risks could occur. In particular, the coe cient r is based on the notions of linearity, normality and mean which are not appropriate when dealing with extreme events. The use of this indicator can lead to underestimation of the risk. Moreover, the Pearson coe cient may not even exist for heavy tailed distributions such as the Generalized Extreme Value or the Generalized Pareto. For instance, in the case of the Cauchy distribution, a theoretical value of Pearson's correlation does not exist. Embrechts et al. [21] showed that the Gaussian model is inadequate to quantify the extreme risks and indicated that the covariance gives incomplete information of joint extreme risks.
The non-parametric dependence measures, Spearman's ρ and Kendall's τ, do not assume linearity and are not based on normality. The Spearman's ρ can be seen as the Pearson's correlation coe cient between the ranked variables [e.g. 61] and measures the average departure from independence [see, e.g. 63, Section B.2.3]. The Kendall's τ is also based on the ranks of the observations [40] and measures the excess of concordance/discordance [see, e.g. 63, Section B.2.2]. These coe cients do not attribute su cient weight to the extreme values. They are good overall indicators but are not appropriate when the focus is on the extremes and the distribution tail.
To study the dependence in the BFA of extreme events, a "local dependence measure" is required since the interest is in the distribution tails. In Extreme Value Analysis (EVA), a number of relevant concepts and methods are developed to locally study the dependence in a joint distribution [e.g. 26,27]. These concepts are commonly used in actuarial sciences and nance [e.g. 1,7,22,50]. For instance, the upper tail dependence parameter is introduced by Joe [37, p. 33]. However, to the best knowledge of the authors, there are no hydrological investigations of such methods for hydrological BFA except the upper and/or lower tail dependence parameter which is, for instance, brie y presented in Salvadori et al. [63], Genest and Favre [29], Poulin et al. [58], Serinaldi [68], Shiau et al. [70] and Lee et al. [46]. Nevertheless, this parameter is not always appropriate and should be combined with other complementary measures.
The aim of the present paper is to introduce and study di erent tail dependence measures for bivariate random variables (X, Y) in hydrological BFA. The paper is organized as follows. In Section 2, we present the recent and signi cant tail dependence measures in EVA. In Section 3, we focus on the special case of Bivariate Extreme Value (BEV) distributions due to their importance in EVA. Non-parametric estimators of the presented tail dependence measures are brie y developed in Section 4. Section 5 is devoted to the applications and Section 6 presents the conclusions.

Tail dependence measures for bivariate distributions
Let (X , Y ), . . . , (Xn , Yn) be independent random vectors in R with joint cumulative distribution function (CDF) F(., .). We denote the marginal distributions of F(., .) as F (.) and F (.) respectively for X and Y and by C(., .) the copula function associated to F(., .). A copula is a cumulative distribution function (CDF) whose margins are uniformly distributed on [ , ]. The joint distribution function can be written in the form [71]: A copula function represents the dependence structure of a multivariate random vector. It contains complete information about the joint distribution apart from its margins. In this sense, a copula describes the association between X and Y in a form that is invariant with respect to strictly increasing marginal transformations [12]. The marginal distributions F (.) and F (.) are assumed to be continuous, which is the case for hydrological series. Therefore the copula C(., .) is unique. The reader is referred to Nelsen [51] or Joe [37] for further details on the theory of copulae and to Salvadori et al. [63] for practical illustrations. In the remainder of the Section, three tail dependence measures are presented. The rst allows to group together the distributions into two classes whereas the second provides a complementary information to that provided by the rst. The third links the rst measure with the second and allows also to reinforce the ndings given by the rst two.
. Tail dependence measure χ U The rst concepts were discussed as far back as Ge roy [26,27] and the following formal de nition has been given by Joe [37, p. 33]: The limit χ U is called the upper tail dependence parameter (UTDP). It roughly corresponds to the probability that one margin exceeds a large quantile threshold u under the condition that the other margin exceeds u as well [24]. In other words, it is the probability that if one variable is extreme, then the other is also extreme.
The formulation in (2) is of interest for hydrological processes, since it is based on F (X) and F (Y) and not directly on X and Y and therefore, the variables describing the hydrological event do not need to have the same scale and to be of the same nature. The UTDP χ U is de ned as the limiting value of χ(u) as u → where [12] Note that in EVA, the statistical study of the tail or the extreme risk is always established under asymptotic considerations. In the remainder of the paper, the term "asymptotic" refers to u → . The function χ(u) can be interpreted as a quantile-dependent measure of dependence [12]. Its upper and lower bounds are given by: The left and right hand sides in (4) correspond respectively to perfect negative and perfect positive dependence [3, p.344]. The function χ(u) provides an insight to the dependence structure at lower quantile levels. The case C(u, u) = u corresponds to exact independence χ(u) ≡ . When χ U ∈ ( , ], then X and Y are said to be asymptotically dependent, whereas when χ U = , these variables are said to be asymptotically independent. In general, χ(u) is a non-trivial function of u and does not have explicit formula. As illustrated in Figure 1a, Coles et al. [12] showed that for a pair of Gaussian variables with correlation coe cient ρ, χ(u) increases with ρ, but as u → the e ect of dependence diminishes and χ(u) → for all ρ < . For an intermediate value of u, χ(u) is reasonably linear with distinctly di erent values for all ρ. For ρ > , χ(u) converges very slowly and ultimately abruptly. An important nding from this Figure is that the dependence in the center is clearly not maintained in the extremes. It is possible to pass from high dependence to independence. On the other hand, this means that it is possible to conclude erroneously that the extremes are asymptotically dependent simply because the extreme independence is not easily detectable due to inadequate sample size. This indicates that the bivariate extreme models are not adapted in the case of asymptotic independence, see, later, the remark after Eq. (13). Therefore, although these models clearly re ect the behaviour of extremes in the case of asymptotic dependence, in the case of asymptotic independence the result is very mixed.
In summary, in the extremes context, although χ U is "better" than overall dependence measures r, ρ and τ, it is not always su cient to quantify the dependence appropriately in all situations. It could fail to discriminate between the degrees of relative strength of dependence for asymptotically independent variables. Thus, it is important to overcome this limitation by introducing another characterization or a complementary dependence measure. Note that χ U is the only measure employed in hydrological applications and it is only considered in few studies [18,32,46]. Before introducing the complementary measure of χ(.) in the next section, it seems to be interesting to focus, brie y, on the auxiliary function qc(.) called the tail concentration function (TCF). It depends on the diagonal section of a copula and is de ned as follows [72] The TCF can be seen as a tool to give a description of tail dependence at nite scale. In addition, it can be more suited to assess the risk of joint extremes than its limits given by χ U and χ L where χ L = lim u→ P(F (X) < u|F (Y) < u). Thus, when the convergence speed of the TCF to is slow, this implies that the dependence in the nite upper tail can be signi cantly stronger than in the limit [19]. The reader is referred to [19,see Fig. 3] for the discussion related to the practical e ect of considering the TCF.

. Tail dependence measureχ U
The function χ(.) given in (3) as a tail dependence measure is useful in the case where the variables are asymptotically dependent. It is not appropriate for discriminating asymptotic independence for which data exhibit positive or negative association, i.e. correlation, that only gradually disappears at more and more extreme levels. A complementary measure of χ(.), denotedχ(.), has been introduced by Ledford and Tawn [43,44] and developed by Coles et al. [12]. The functionχ(.) measures the strength of dependence within the class of asymptotically independent distributions. In a similar way to the function χ(u) given in (3),χ(u) is de ned as followsχ whereC(u, v) = − u − v + C(u, v). The functionχ(u) is also bounded from below and above as log( − u) log(max( − u, )) − ≤χ(u) ≤ , < u < .
χ(u) has the following properties [2,3,12]: • If an exact independence occurs beyond u, thenχ(u) = ; • If there is a perfect dependence beyond u, thenχ(u) = ; • Ifχ(u) ∈ ( , ), then P F (X) > u|F (Y) > u > P F (Y) > u and the extremes are positively associated; i.e. observations for which both F (X) > u and F (Y) > u for large threshold u are likely to occur more frequently than under exact independence between X and Y; • Ifχ(u) ∈ (− , ), then P F (X) > u|F (Y) > u < P F (Y) > u and we say that the extremes are negatively associated, i.e. observations for which both F (X) > u and F (Y) > u for a large threshold u are likely to occur less frequently than under exact independence between X and Y; • |χ(u)| increases with the tail dependence.
To focus on extremal characteristics, by analogy to χ U , one de nesχ U as the limiting value ofχ(u) as u → for which − ≤χ U ≤ . This limit has the following properties: •χ U = corresponds to the asymptotic dependence of extremes. The bivariate Gumbel-logistic extreme value distribution is an example where this case occurs; •χ U < corresponds to the asymptotic independence of extremes andχ U provides a limiting measure that increases with relative dependence strength within this class; •χ U allows to better characterize a possible asymptotic independence and it provides a complementary information to that provided by χ U . For instance, as illustrated in Figure 1b, in the case of a Gaussian pair, we haveχ U = ρ andχ(u) is approximately linear for . < u < . Therefore, one concludes an asymptotic independence, despite what might suggest a direct interpretation of χ(u) in Figure 1a [12].
In summary, the quantities χ U andχ U allow to characterize the dependence of extremes as follows: • χ U ∈ [ , ] with the set ( , ] corresponds to asymptotic dependence; •χ U ∈ [− , ] with the set [− , ) corresponds to asymptotic independence.
As a result, the pair (χ U ,χ U ) can be used as a summary of extreme dependence: • If (χ U > ,χ U = ), the variables are asymptotically dependent and χ U determines a measure of strength of dependence within the class of asymptotically dependent distributions; • The case (χ U = ,χ U < ) corresponds to asymptotic independence between variables andχ U measures the strength of dependence within the class of asymptotically independent distributions.

. Coe cient of tail dependence η
In this subsection, we assume that a joint distribution of (X, Y) has unit Fréchet margins, i.e.
This restrictive assumption is without loss of generality since, if necessary, F (.) and F (.) can be transformed into unit Fréchet margins under suitable assumptions [see e.g . 43]. In order to analyse the asymptomatic dependence structure between the Fréchet margins and to link χ U withχ U , Ledford and Tawn [43,44] introduced the following model on the tail of the joint survival function of (X, Y): where L is a univariate slowly varying function at in nity [5, Theorem 1.5.12], i.e. , The rate of decay in (10) is primarily controlled by η. The coe cient η describes the type of limiting dependence between X and Y, and L is its relative strength given a particular value of η. By putting T = min(X, Y), it follows that P(X > z, Y > z) = P(T > z) ∼ z − /η L(z) and η is identi ed as the tail index of the variable T. Hence, the usual univariate techniques can be used to evaluate η [35,55]. One can show that and the estimate ofχ U can be obtained from that of η which is more developed and studied since it is related to the tail index. As a consequence, we have [12,34,43,45]: • X and Y are asymptotically dependent if and only if η = and L(z) → c ∈ ( , ] as z → ∞. In this situation, we have (χ U = c,χ U = ). The constant c denotes the dependence degree where c = corresponds to the perfect dependence in tail; • The case η → and L(z) = corresponds to perfect negative dependence (in tail); In addition, within the class of asymptotically independent variables, i.e. < η < , three types of independence can be identi ed: • The case η = / corresponds to near independence between the extremes of X and Y. These extremes are exactly independent when c = ; • If / < η < and c > , or η = and c = , then the marginal variables are said to be positively associated; • If < η < / , then the marginal variables are said to be negatively associated.
To summarize, the degree of dependence between large values of Fréchet margins is determined by η, with increasing values of η corresponding to stonger association. For a given η, the relative dependence strength is characterized by the slowly varying function L [3, p.346]. For instance, for the Gaussian dependence model with correlation ρ < illustrated in Figure 1, [34]. In that case, positive association, negative association and exact independence arise respectively as ρ > , ρ < and ρ = . The perfect positive and negative associations are reached as ρ → and ρ → − respectively. Figure 2 summarizes in a diagram the presented tail dependence measures by highlighting the concepts of the asymptotic independence/dependence. Figure 2 gives also additional information which is developed in the following Section. In Figure 2, the circle denotes the starting point, with several possible paths that can be followed. This Figure will be described later, at the end of Section 3.

Particular case of the BEV distributions
The BEV distributions are a particular case of bivariate distributions. They are characterized by some speci c dependence functions which can be expressed through the previous tail dependence measures. In this Section, we brie y present the relevant measures of the tail dependence for these distributions since they play a prominent role in the studies of bivariate extreme events. In order to carry out a meaningful study about tail dependence in the BFA, we assume that F(., .) belongs to the domain of attraction of a BEV distribution G, i.e. there exist standardizing sequences an , cn > and bn , dn ∈ R such that for all x and y [25,60] It is shown in the literature that all BEV distributions are asymptotically dependent, otherwise, in the case of an asymptotic independence, the only possible situation is the exact independence [e.g. 2,12]. For the latter,χ U = and χ U > , and in practice the dependency is the stronger as the UTDP χ U is close to . Besides, the tail dependence function χ(.) is constant. Figure 3 illustrates the behaviour of the tail dependence functions χ(.) andχ(.) for the bivariate Gumbel-logistic distribution with dependence parameter < θ ≤ which is a BEV distribution. Notice that, for the bivariate Gumbel-logistic distribution, the parameter θ measures the strength of the dependence and the limiting cases θ = and θ = correspond respectively to independence and perfect dependence. Figure 3a shows that χ(.) is positive, constant and close to when θ is close to . In Figure 3b, for large values of θ,χ(.) converges slowly to as u → .
An estimation of χ(.) signi cantly non-constant re ects an inadequacy of the BEV distribution to the data. This situation arises when (X, Y) are asymptotically independent and n, the block size maxima, is not large enough to meet the condition in (13) [12]. In hydrological BFA, since the peak ows are extracted as Bivariate tests of extreme-value dependence (a) and (b) or g.o.f. test (c) for bivariate extreme-value copulae presented in end-subsection 3.1.

BEV distributions and Domain of Attractions:
X and Y are asymptotically dependent of degree c.
Perfect negative dependence.
The class of asymptotically dependent variables Consider model (10) and evaluate η.
Evaluate the tail coe cients χ U and χ U respectively via Eqs. (3) and (7). Negative association: observations for which X and Y exceed a large threshold z occur less frequently than under exact independence.
Positive association: observations for which X and Y exceed a large threshold z occur more frequently than under exact independence.
Extremes of X and Y are near independent.
There is exact independence when L(z) = . block maxima, hydrologists tend to jointly model ood characteristics with the component-wise maxima, i.e. a BEV distribution, without always checking rst ifχ U = . The dependence structure of G(., .) in (13) is characterized by quantities given in the following subsections.

. Pickands dependence function
The representation of dependence structure discovered by Pickands [56] turned out to be far more convenient than its predecessors [3, p. 270] such that: where and De nition (15) implies the following relationship between χ U and A(.): Thus, estimating χ U is a particular case of estimating A(.). The reader is referred to Salvadori and De Michele [62] for practical applications of Pickands dependence function in hydrology. The Pickands dependence function A(.) and the extreme value copula C(., .) allow to check whether a sample comes from a BEV distribution G(., .) or at least if F(., .) belongs to the domain of attraction of a BEV distribution G. To this end, three statistical tests can be used: (i) the bivariate test of extreme-value dependence based on Kendall's process [33] or (ii) the one based on the Pickands dependence function [41] and then, if H is accepted, (iii) the goodness-of-t tests for bivariate extreme-value copulae [30] .

. Stable tail dependence function
A bivariate CDF F(., .) with continuous margins F (.) and F (.) is said to have a stable dependence function (STDF) (., .) if the following limit exists [36]: Referring to [17], F(., .) is equivalent to (13) if and only if (i) F (.) and F (.) are in the max-domains of attractions of extreme value distributions G (.) and G (.) respectively, and (ii) F(., .) has a STDF (., .) de ned by where G(.) is a BEV distribution while γ and γ are real constants called the marginal extreme value indices. The STDF (., .) can be seen as a starting point to construct non-parametric models or BEV distributions. For instance, one can cite the Gumbel-logistic model for which (x, y; θ) = x /θ + y /θ θ , x, y ≥ and < θ ≤ . The Recall that for BEV distributionsχ U = . .

Tail copula function
The tail copula is a function that describes the dependence structure in the tail of a joint CDF F(., .). Similar to (17), for all non-negative x and y, the quantity is called the tail copula function (TCF) of (X, Y), provided the limit exists. The relationship between Λ(., .) and (., .) is given by The quantity Λ( , ) is the UTDP χ U of (X, Y) [e.g. 20]. Schmidt and Stadtmuller [65] proposed Λ(., .) as a starting point to construct a multivariate distribution of extreme values. In addition, the TCF function is considered as an intuitive and straightforward generalization of the tail dependence function χ(.) via a function describing the dependence structure in the tail of a distribution [65]. To summarize, in these particular cases, χ U is expressed explicitly with Λ(., .), (., .) and A(.) as follows: These relations are useful for estimating χ U since the established properties of the functions Λ(., .), (., .) and A(.) are well developed. Figure 2 summarizes in a diagram all presented tail dependence measures, χ U ,χ U , η, Λ(., .), (., .) and A(.), by highlighting the concepts of the asymptotic independence and asymptotic dependence. In Figure 2, from the starting point circle, there are several possible paths ((A), (B ), (B ) and (C)) which can be followed. The choice of which path to take depends on the available information and the goal. However we recommend to follow the path (A). It can be seen as a procedure starting from data to obtain the tail behaviour via the presented measures. The path (A) describes as follows: (i) From the bivariate data {(X i , Y i ), i = , . . . , n} with joint distribution function F(., .), evaluate the tail coe cients χ U andχ U respectively given by (3) and (7). If (χ U > ,χ U = ), we are within the class of asymptotically dependent distributions; otherwise if (χ U = ,χ U < ), we are within the class of asymptotically independent distributions. (ii) Consider model (10) and evaluate η: (1) Within the class of asymptotically independent distributions, depending on the values of η, four cases are possible: negative association, positive association, near independence or exact independence. (2) Within the class of asymptotically dependent distributions, according to η, three types of dependence are possible: perfect negative dependence, perfect positive dependence and asymptotic dependence. (iii) Except for the exact independence, all BEV distributions are asymptotically dependent.

Non-parametric estimation of tail dependence
Depending on the level of available information about the distribution of the data, there exist several approaches to estimate the tail dependence functions and coe cients. First, the bivariate distribution F(., .) could be either known [22] or belongs to a class of distributions [64,65]. Second, the tail dependence can be estimated by using a speci c copula [49,53] or a class of copulae [39]. Finally, non-parametric estimation methods can be employed when no speci c form is known or constrained on the copula or on the marginal distributions. In the present section, we focus on non-parametric methods. The tail dependence estimates are obtained from the empirical copula or based on the transformation of original data to Fréchet variables because F(., .) or C(., .) are generally unknown [57].

. Estimators of tail dependence parameter χ U
As shown implicitly in Section 2, the tail dependence parameter χ U can be estimated by using the copula, the Pickands dependence function, the STDF or the TCF. In the following, one shows how an estimator of χ U is obtained by using the latter functions.

. . Estimation via the empirical copula
An estimator of χ(.) is obtained via the empirical copula. [38] introduced the following estimator where k denotes a threshold, that is a sample fraction, to be chosen andĈn(., .) is the empirical copula de ned by [14,31] where 1 {.} is the indicator function, while R X i and R Y i , respectively, stand for the ranks of X i among X i , . . . , Xn and Y i among Y i , . . . , Yn. Coles et al. [12] introduced, on the basis of (3), the following estimator of the tail dependence function whereĈm(., .) is an empirical copula computed from m block maxima X * lj and Y * lj , j = , . . . , m, and where each block contains l = n/m elements of the original data. The estimatorsχ SEC U andχ LOG U are deduced respectively fromχ SEC (.) andχ LOG (.) by noting that u = (n − k)/n is close to when k is small.
The coe cient χ U can also be estimated by the least-square method such that [15,23]: where arg min λ∈ [ , ] h(λ) gives an argument at which h(.) is minimized over the domain [ , ]. Dobric and Schmid [15] showed that k ≈ √ n can be an appropriate choice to built the estimatorsχ SEC U ,χ LOG U andχ FD U . Frahm et al. [24] suggested to deduceχ SEC U andχ LOG U by choosing a threshold k based on the property of tail copula homogeneity as stated in Schmidt and Stadtmuller [65,Theorem 1]. This approach consists in identifying a plateau, which is induced by the homogeneity, on the graphs (k,χ • (.)). Nevertheless, the plateau-nding algorithm developed in Frahm et al. [24] requires a prior de nition of some parameters.

. . Estimation via Pickands dependence function A(.)
As mentioned in subsection 3.1, estimating χ U can be obtained by estimating A(.) via (16). Since the margins F (.) and F (.) are rarely known in practice, a natural way to proceed is then to estimate them empirically byF ,n (.) andF ,n (.). This leads to estimating the copula C(., .) on the basis of the transformed observations {(F ,n (X i ),F ,n (Y i )), i = , . . . , n}. However, it is more convenient to consider scaled variables de ned by Genest and Segers [31]: The scaled pairs {(Û i ,V i ), i = , . . . , n} are called the pseudo-observations from copula C(., .). They allow to avoid dealing with points at the boundary of the unit square. Genest and Segers [31] proposed the two following estimators of A(.) which are the rank-based versions of the estimators given respectively by Pickands [56] and Capéraà et al. [8]: where c E ≈ . is the Euler's constant while, for i ∈ { , . . . , n}, the functionξ (.) is de ned asξ i ( ) = − logÛ i ,ξ i ( ) = − logV i and for all t ∈ ( , )ξ i (t) = min − logÛ i −t , − logV i t . The estimators in (29) lead tô χ P U ,r = − Â P n,r ( / ) andχ CFG U ,r = − Â CFG n,r ( / ). Another estimator of the UTDP, motivated by Capéraà et al. [8] and studied by Frahm et al. [24], is given bŷ The latter estimator relies on the hypothesis that the underlying empirical copula can be approximated by an extreme value copula. As the margins are unknown, in practice one can replace F (X i ) and F (Y i ) by the scaled variablesÛ i andV i . Note that, in some situations, the Pickands and the CFG estimators can be altered to meet the endpoint constraints A( ) = A( ) = . Therefore, for all t ∈ [ , ], Segers [67] suggested endpoint-correction versions of the Pickands and CFG estimators. Genest and Segers [31] showed that the endpoint correction to estimators (29) has no impact on their limiting distributions. In addition, they showed that that the CFG estimator is generally preferable to the Pickands one when the endpoint corrections are applied to both of them.
The distribution ofˆ H n (., .) is independent of the continuous margins [17]. Einmahl et al. [20] proposed the following estimator that usually performs slightly better thanˆ H n (., .) for nite samples: where R X i and R Y i as in (25). De nitions (31) and (32)

. Estimators of tail dependence parameterχ U
Estimator ofχ U can be obtained on the basis of the copula or the tail coe cient η. The functionχ(.) can be estimated by substituting in (7) the empirical estimate of survival copula functionĈn(., .) given byĈn(u is given in (25). Thus, a non-parametric estimator ofχ(.) is given by [12] On the other hand, according to (12) an estimation of η leads to an estimation ofχ U . Since η is identi ed as the tail index of the univariate variable T = min(X, Y), one can estimate η with the estimator called Zipf [42,66] where T ,n ≤ . . . ≤ Tn,n denote the order statistics of the random variables T i . It can also be estimated with the Hill [35] estimator given by: The two latter estimating procedures require the knowledge of the margins F (.) and F (.) since the model of Ledford and Tawn [43,44] assumes that (X, Y) has unit Fréchet margins. However, when the margins are not identical or not Fréchet distributed, the original variables can be transformed to standard Fréchet margins de ned by Xnew = − / logF (X original ) and Ynew = − / logF (Y original ) [43]. However, these transformations could induce an uncertainty in the estimates of η [3, p.351]. Therefore, Peng [54] and Draisma et al. [16] respectively proposed non-parametric alternatives (36) and (37) based directly on the empirical distributions of the original observations given respectively by: where Sn(k) = n i= 1 {X i >X n−k+ ,n , Y i >Y n−k+ ,n } with k = , . . . , n − . For the choice of the threshold k, the reader is referred to Lekina et al. [48] and references therein.

Application to floods
In this Section, the presented estimators of the tail dependence are applied to a particular hydrological event, namely oods. Flood events are mainly described by three characteristics that are ood peak (Q), ood volume (V) and ood duration (D).

. Data description
The data used in this case study consists in daily natural stream ow measurements from three stations in the province of Quebec (Canada). The reference numbers of the selected basins are , and and the gauging stations are respectively denoted by ST , ST and ST . Maximum annual ood events are described by their ood peaks, durations and volumes as extracted from the daily stream ow data.The three variables correspond to the same ood event each year. In particular, it corresponds to the spring ood event which is the important ood event in Quebec and is caused mainly by snow melting [4]. Gauging

Station Localisation
Latitude Longitude Observation years Batiscan River The hydrological literature has highlighted issues concerning the correlation between the three characteristics of ood events. Due to space limitations, the focus will be made on the study of (Q, V) whereas brief results will be provided concerning (V , D) and (Q, D). The couple (Q, V) is generally highly correlated and represents the most studied in the literature [see e.g. 11,69,74,75]. The tail dependence behaviours of the couples (Q, V), (D, V) and (V , D) are studied according to the proposed approach summarized in Figure 2. This step is performed before any modelling of the joint distributions. While in Chebana and Ouarda [10] the bivariate descriptives statistics based on the depth function are investigated, the present study focuses solely on the study of the tail dependance. Before presenting the results, it should be noted that the hydrological analysis is generally a ected by the record length especially when we deal with extremes. Despite this limitation, asymptotic results are usually employed in the hydrological frequency analysis and in the multivariate setting in particular. This issue could have similar e ect on the tail dependence measures. However, it is expected that in the future data will be more available and hence this issue would have less and less impact.

. Tail dependence measures
Since the distributions of the series (Q, V), (Q, D) and (V , D) are unknown, the tail dependence function χ(.) and its complementary functionχ(.) are directly evaluated via the empirical copula. This allows to assess the strength of tail dependence.
The estimatorsχ SEC (.) (with m = n),χ LOG (.) andχ FD U , de ned respectively in (24), (26) and (27), are evaluated. First,χ SEC U andχ LOG U are deduced respectively from the functionsχ SEC (.) andχ LOG (.) by noting that u is close to for small k. Second, the UTDP estimatorsχ SEC U ,χ LOG U andχ FD U are obtained by xing k = √ n. The coe cientχ U is deduced from the functionχ(.) which is estimated via the empirical survival copula as well as via the coe cient of tail dependence η. Hence, one uses respectively the estimatorχ COLES de ned in (33) and the estimatorχ • U ,k = η • k − whereη • k is de ned in (34), (35) or (37). The symbol • denotes one of the indices Z, H or D. Since the margins are unknown, the estimatorsη Z k andη H k , given in (34) and (35) respectively, are computed by rst transforming the margins to standard Fréchet margins. The threshold k is chosen in the simultaneous stability range of the estimatorsχ H U ,k ,χ Z U ,k andχ D U ,k and the corresponding estimator is denotedχ * U . This technique is commonly used for the estimation of the tail index or extreme quantile in EVA [e.g. 48]. The overall estimated dependence coe cients of Pearson's, Kendall's and Spearman's are denoted respectively byrn,τn andρn are evaluated for comparison purposes.

Tail dependence for the series (Q, V)
In the remainder of the Section, the analysis is presented rst according to gauging stations and then according to the measures of dependence. Figures 4 and 5 illustrate the di erent estimators of tail dependence functions χ(.) andχ(.) respectively.
Generally, Figure 4 shows that the estimators of χ(.) are not too close to when u is large enough. Therefore the tail dependence is not strong in the three considered stations. Since the degree of dependence is not strong, this suggests to restrict the analysis to the univariate case only. However, as it will be seen in the obtained results by an in-depth analysis in the remainder of this section, proceeding in this manner may be inappropriate or misleading.
For ST , (Figure 4a), all estimators of χ(u) are considerably larger than for u < . . When u is close to , i.e. u > . , the estimatorsχ LOG (u),χ SEC (u) and the estimated UTDPχ DF U converge to abruptly with respect to u or k and the di erence between these estimators andχ DF U is large. It would have been possible to conclude erroneously that the couple (Q, V) is asymptotically dependent in station ST . Similar results are obtained by [12] for other data. In addition, for u = − k/n ≈ .
. Figure 5a indicates that all estimators ofχ U are signi cantly di erent from . For instance, they are almost stable around the interval [ .
for u = − k/n ≈ . . Indeed, according to the properties of the coe cientχ U given in subsection 2.2 and summarized in Figure 2, the peak ow and the ood volume of ST can not be described by BEV distributions since there is no asymptotic dependence. As indicated by the paths (A) or (B ) in Figure 2, we recall that in ST , an analysis based only on the estimators of χ U does not guarantee that the couple (Q, V) is asymptotically independent. Figure 5a indicates that: ),χ LOG ∈ [ , .
) andχ SEC ∈ [ , . ) for . < u < . This leads to conclude that the extremes are positively associated, i.e. in ST the observations for which both F (Q) > u and F (V) > u for large thresholds u occur more frequently than under exact independence between Q and V.
≤ u < , we haveχ LOG ∈ ( , . ],χ SEC ∈ ( , . ] andχ DF ∈ ( , . ]. Accordingly, with respect to path (A) in Figure 2, the couple (Q, V) seems to be asymptotically dependent. On the other hand, Figure 5b indicates that the estimatorsχ H U ,k andχ Z U ,k have a regular behavior and are almost stable for large thresholds. This indicates more accurate evaluation ofχ. In other respects, the tail dependence estimatorsχ COLES andχ • U ,k are non-negative for < u < and < k < n respectively. More precisely, χ COLES U = .
for u ≈ . whereas the estimatorsχ • U ,k are located in the range [ . , ]. In particular, χ D U ,k uctuates slightly around u ≈ and the estimatorsχ H U ,k andχ Z U ,k are almost stable and approximately equal to . This suggests thatχ * U ≈ and one can then conclude that the couple (Q, V) is asymptotically dependent, see path (B ) in Figure 2. In additionχ U ∈ [ .
The estimated overall dependence coe cients, the estimated tail dependence parameters and the estimated tail dependence functions for u ∈ ( − k/n, ) are summarized in Table 2. Table 2 shows that, in all three stations, the overall coe cients lead to conclude that there are signi cant correlations which are not very high. Nevertheless, as concluded previously on the basis of tail dependence measures, in ST the extremes are asymptotically independent. Thus, one observes that an analysis solely based on the overall dependence coe cients does not give enough information to re ect the nature of the relationship between extremes of the couple (Q, V) in ST , ST and ST . As previously mentioned (see Section 3), all BEV distributions are asymptotically dependent. Since it was concluded that the couple (Q, V) in ST is asymptotically dependent, one of the BEV distributions could be a candidate for the sample of (Q, V), for instance, an extreme value copula or the Gumbel-Hougaard family. The bivariate tests presented in subsection 3.1 are used to check this. Results are provided in Table 3. The Gumbel-Hougaard copula family is commonly used for hydrological FA [e.g. 32]. Notice that for a given degree of dependence, the most popular extreme value copulae are strikingly similar [e.g. 30]. The tests require estimating the Pickands dependence function A(.). For the Ghoudi et al. [33] bivariate test, the approximative p-values obtained by jackknife, the nite sample plug-in and the asymptotic plug-in are noted pv.jac, pv.fsa and pv.asy respectively. For the Kojadinovic and Yan [41] bivariate test, the p-value is denoted by pv.ky. For the Genest et al. [30] goodness-of-t tests, pv.mpl, pv.itau and pv.irho are the approximative p-values obtained respectively by using parametric bootstrap combined with the maximum pseudo-likelihood, the method of the inversion of Kendall's tau and the method of the inversion of Spearman's rho. In Figure 6, we present for t ∈ [ , ] the rank-based estimatorŝ A P n,r (t) andÂ CFG n,r (t) de ned in (29), and the corresponding corrected endpoint estimators notedÂ P n,c (t) and A CFG n,c (t) [67]. The estimated UTDPχ P U andχ CFG U which are related, via (16), to the dependence function A(.) and the obtained p-values of all bivariate statistical tests of extreme value dependence are summarized in Table 3. Notice that an analysis based on Pickands dependence function or the UTDP estimators lead to the same ndings.
The analysis on tail dependence function χ(.) and its complementary functionχ(.) allow to conclude that the couple (Q, V) is asymptotically independent for ST . To consolidate this nding, the function A(.) is estimated and the p-values of the bivariate tests used previously are computed. Figure 6a suggests  , . ] which are lower than . Nevertheless, as shown in the previous analysis based on tail dependence function χ(.) and its complementary functionχ(.), this represents only a graphical indication. In fact, the bivariate statistical tests in Table 3 con rm that we can not model the couple (Q, V) by a BEV distribution and especially by the Gumbel-Hougaard family copula since pv.jac, pv.fsa, pv.asy, pv.ky, pv.mpl and pv.itau are lower than . . Figure 6b indicates that in ST , Q and V are asymptotically dependent since via the relationship (16) Table 3 are higher than . which con rms a good t with the BEV distributions. Then, for ST , the dependence of (Q, V) can be modelled with the Gumbel-Hougaard family copula. In addition, fromχ P U andχ CFG U , one deduces that the degree of dependence between Q and V is within the interval [ .
]. Notice that even though this degree of dependence is slightly lower than the previous values, i.e.χ U ∈ [ .
, . ], where no assumption on the model was made, the same conclusion is obtained: i.e. there is asymptotic dependence.
In Figure 6c, the indication graph suggests an asymptotic dependence between Q and V in ST ] are lower than . Moreover the bivariate statistical tests in Table 3 con rm this graphical indication since the obtained p-values are higher than . . However, this nding is not compatible with this resultχ * U = .
≠ which means that there is asymptotic independence. This could be explained on the basis of construction of the tests used. Indeed, the tests used are based only on the function A(.) and not on the tail-dependence measureχ(.). In addition, the p-value is a measure of the evidence against the null hypothesis: the smaller the p-value, the stronger the evidence against the null hypothesis. A large p-value is not strong evidence in favour of null hypothesis. A large p-value can occur for two reasons: (i) null hypothesis is true or (ii) null hypothesis is false but the test has low power. The p-value is not the probability that the null hypothesis is true [see 73, p.157].

Tail dependence for the series (Q, D) and (V , D)
In  Table 4. Figures 7a, b, Figures 8a, b and Table 4 show that Q and D are asymptotically independent in both ST and ST . More precisely, in ST we observe near independence sinceχ ≈ andχ * U ≈ whereasχ COLES ≈ for < u < .
andχ COLES = − for . ≤ u < . Figures 7c and 8c indicate that Q and D are asymptotically independent in ST sinceχ ≈ . Based on the estimatorsχ D U ,k andχ COLES , one can deduce that the association between Q and D is negative. Otherwise, one can deduce a positive association when the analysis is only based on the estimatorsχ H U ,k andχ Z U ,k . Nevertheless following Figure 2, an analysis of Figures 7d and 8d does not allow to conclude for an asymptotic independence between V and D in ST since the estimators of χ(.) are clearly inside ( , ) (see also Table 4).
≤ u < . Figures 7e, 8e and Table 4 suggest that V and D are asymptotically dependent in ST . Figures 7f and 8f do not allow to conclude for an asymptotic dependence between V and D in ST since the estimators of χ(.) converge to values around .
, the estimators of χ(.) converge abruptly to . In addition, as the UTDP estimators are in the range ( . , . ) for u ≈ . (see Table 4), one might conclude at best for an asymptotic independence with positive association since the estimators ofχ(.) are non-negative for u ∈ ( . , . ).
In Table 4, the obtained estimated overall coe cients (rn ,τn ,ρn) lead simply to conclude that a significant overall correlation exists in all cases. This shows once again that the hydrological analyses based on overall coe cients are inadequate to quantify the extreme risks that occur at the tail of the distribution. More- over, this case study shows that the measure χ(.) alone is not always su cient to exhibit the relationship between the extremes. Notice that the overall coe cients for the couple (Q, D) in ST are negative. This could indicate that the couple (Q, D) is in fact negatively associated which is not in accordance with the last nding based on tail dependence measures where the sign of the association was not clear. The summarized results in Table 5 show that the couple (Q, V) is asymptotically independent and positively associated in the Bastican River (ST ). In ST , similar to the Gaussian dependence model studied by Coles et al. [12], one can not conclude using only χ U . On the other hand, the overall coe cient values indicate that (Q,V) are relatively highly correlated. However, the conclusion on the tail is not the same where it is shown that the couple (Q,V) for this station is asymptotically independent. In the Harri-