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Depend. Model. 2021; 9:179–198 Research Article Open Access Cécile Mercadier* and Paul Ressel Hoeding–Sobol decomposition of homogeneous co-survival functions: from Choquet representation to extreme value theory application https://doi.org/10.1515/demo-2021-0108 Received April 18, 2021; accepted August 26, 2021 Abstract: The paper investigates the Hoeding–Sobol decomposition of homogeneous co-survival functions. For this class, the Choquet representation is transferred to the terms of the functional decomposition, and in addition to their individual variances, or to the superset combinations of those. The domain of integration in the resulting formulae is reduced in comparison with the already known expressions. When the function under study is the stable tail dependence function of a random vector, ranking these superset indices corre- sponds to clustering the components of the random vector with respect to their asymptotic dependence. Their Choquet representation is the main ingredient in deriving a sharp upper bound for the quantities involved in the tail dependograph, a graph in extreme value theory that summarizes asymptotic dependence. Keywords: Hoeding–Sobol decomposition, co-survival function, spectral representation, stable tail depen- dence function, multivariate extreme value modeling. MSC: 26A48, 26B99, 44A30, 62G32, 62H05 1 Introduction d 2 d Let f : [0, 1] ! R be a function in L ([0, 1] , λ) where λ = λ is a product of probability measures i=1 on [0, 1]. One way to understand the structural form of the d-variables function f is to decompose it into func- tions of increasing complexity. This is precisely what allows the functional analysis of variance (FANOVA). It relies on the Hoeding–Sobol decomposition f(x) = f (x) (1) uf1,...,dg where Z ju\vj f (x) = (−1) f(x)dλ (x) (2) u −v vu for dλ (x) = dλ (x ) and−v = f1, . . . , dg\v. See [5, 18, 19]. The term f only depends on the components u u i i i2u R R of x associated with u. The constant term f is equal to fdλ and the global variance is given by σ = (f − 2 2 2 2 f ) dλ. Set σ = f dλ and σ = 0. Then, from orthogonality arguments (see, for instance, [2]), the term ? u u u ? f is centered (except for the empty set) and the FANOVA expression relies on the equality 2 2 σ = σ . uf1,...,dg *Corresponding Author: Cécile Mercadier: Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan, UMR CNRS 5208, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France, E-mail: mercadier@math.univ-lyon1.fr Paul Ressel: Kath. Universität Eichstätt-Ingolstadt, Ostenstraße 26-28, 85072 Eichstätt, Germany, E-mail: paul.ressel@ku.de Open Access. © 2021 Cécile Mercadier and Paul Ressel, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 180 Ë Cécile Mercadier and Paul Ressel 2 2 2 Interest in the individual variances σ , and more particularly their ratio to the total variance σ /σ , traces u u back to [18] and [12]. The current research problems in Global Sensitivity Analysis (GSA) are varied in nature. Our concern in this paper is not improvements for estimation, cost-saving, construction of surrogate models, or other practical but no less crucial aspects or perspectives ; see rather [11, 13] and references contained therein for an overall and recent assessment. The main goal here is to reveal simplied theoretical expres- 2 2 sions for the quantity σ within a specic class of functions. Knowing such quantities σ allows to order the u u importance of the input variables x , . . . , x with respect to the global variance of f , the function under study. 1 d Reducing the number of variables of interest in f is one of the main consequences of this hierarchical ranking. In this paper we will concentrate on homogeneous co-survival functions. Classical examples to keep in 1/t t d mind are the power norms, dened for t ≥ 1 by ψ (x) := x for x 2 [0,∞[ . More generally, if there i=1 i d d exists μ a non-negative Radon measure on [0,∞] \ f∞g such that ψ(x) = μ(fy 2 [0,∞] jy ̸≥ xg) for all d d x 2 R then ψ : R ! R is said to be a co-survival function. The class of co-survival functions additionally + + assumed to be homogeneous, is in a one-one correspondence (modulo value at (1, . . . , 1)) with probability measures ν on C = fw = (w , . . . , w ) 2 [0, 1] j max(w , . . . , w ) = 1g. Indeed, the spectral representation 1 d 1 d ψ(x , . . . , x ) = ψ(1, . . . , 1) max(x w , . . . , x w )dν(w) (3) 1 d 1 1 d d is stated in [16, Theorem 2]. Some details are given in Section 2 to make the paper almost self-contained. In extreme value theory, stable tail dependence functions (stdf), usually denoted by `, play a central role to describe the asymptotic dependence between components of a random vector X = (X , . . . , X ). Assuming 1 d the existence of a multivariate domain of attraction for the componentwise maxima of X is a classical starting point. This is equivalently written as −1 −1 lim t 1 − F(F (1 − x /t), . . . , F (1 − x /t)) = `(x , . . . , x ) 1 1 1 d d d t!∞ in terms of F, F , . . . , F the cumulative distribution functions of X, X , . . . , X . More details on multivariate 1 d 1 d extreme value theory can be found, e.g., in [1, 3, 4, 7, 10]. As pointed out in [16], the stdfs are particular cases of homogeneous co-survival functions. The corresponding probability measures ν in (3) must satisfy d constraints induced by the fact that a stdf equals 1 at unit vectors. A graph based on the Hoeding–Sobol decomposition of a stdf, called the tail dependograph, has been introduced in [10]. It reveals the asymptotic dependence structure of the random vector X through the structural analysis of the function `. Tail superset indices, which are the superset combination of individual variances, are of prime interest in the tail depen- dograph. Their pairwise values dene the thickness of the edges. The aim of this paper is twofold. On the one hand, we shall establish a simplied expression for the individual variances σ , as for their superset combinations, when the function under study is a homogeneous co-survival function. Their resulting Choquet representation thus provide new test cases for GSA. On the other hand, we will apply these results to stdfs so that upper bounds for the tail superset indices will be obtained. Proving this majorization initially motivated the current study. The paper is organized as follows. We rst investigate the class of homogeneous co-survival functions: in Section 2, the expression of the FANOVA eect ψ and the corresponding variance σ are written as integrals of rank-one tensors (which are products of univariate functions in each of the input parameters, as dened by [8]). The numerical performance of our results is analyzed at the end of this part. As an application, the study focuses on stdfs in Section 3. The new expressions allow to derive some sharp upper bounds for the tail superset indices. All proofs are postponed to Section 4. Finally, the last lines summarize conclusions and references. Notation. Let _ and ^ stand respectively for the maximum and the minimum. Set x = x _ 0. The indicator c d u 1 equals 1 on A and 0 on A . Set 1 = (1, . . . , 1) 2 R . The vector z is the concatenation of z for i 2 u so A i Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 181 P P u −u that (z , x ) = z e + x e in the canonical basis (e , . . . , e ). Binary operations are understood i i i i 1 d i2u i2̸u componentwise, e.g. x · w = (x w , . . . , x w ), x _ w = (x _ w , . . . , x _ w ), s/w = (s/w , . . . , s/w ) 1 1 d d 1 1 d d 1 d Q Q c d and 1 = 1 for s 2 [0, 1]. Throughout the paper [x,∞] := fy 2 [0,∞] jy ̸≥ xg. Let λ = λ s≥x s≥x u i i2u i i=1 be an arbitrary product of probability measures on [0, 1]. For positive v and w, when s and t lie in [0, 1], let K (w, v; s, t) = λ ([0, (s/w)^ (t/v)^ 1]) = 1 dλ (x) and let K (w; s) stand for K (w, w; s, s). The i i i i i (s/w)^(t/v)≥x notation ψ is used for a homogeneous co-survival function whereas ` represents a stdf. 2 FANOVA of homogeneous co-survival function In this section, the functional decomposition is explored under a new setting by considering homogeneous co-survival functions. Before stating our main result, we give a description of the class under study. It is worth noticing that focusing on the unit hypercube [0, 1] is not restrictive by homogeneity assumption. 2.1 Choquet representation of homogeneous co-survival functions Similar to distribution functions, also co-survival functions are essentially characterized by a special mul- tivariate monotonicity property. First, we introduce a notation. Let A , . . . , A be non-empty sets, A = 1 d A × ··· × A , and let f : A ! R be any function. Then for x, z 2 A we put 1 d x juj u −u D f := (−1) f(z , x ) . uf1,...,dg Q Q −u Moreover, for a non-empty subset u ( f1, . . . , dg and for x 2 A , let us dene on A j j j2−u j2u −u u u −u f(·, x )(z ) := f(z , x ) . If A R for all j, the function f is called 1 −alternating¹ if D f ≤ 0 for x ≤ z (both in A), and if this inequality j d z also holds whenever some of the variables are xed, for the function of the remaining variables, i.e. if for each Q Q v v non-empty subset v ( f1, . . . , dg, for each y 2 A and any x ≤ z both in A , we have j j j2−v j2v D v f(·, y) ≤ 0 . See [17] for a detailed presentation of this concept. d d Let f : R ! R be the co-survival function of μ, a non-negative Radon measure on [0,∞] \f∞g, i.e. for any x 2 R f(x) = μ([x,∞] ) . If the reader is not familiar with Radon measures, one should only keep in mind that this assumption ensures that f is well dened and f(x) nite for any x 2 R . By Theorem 3 in [16] one knows that it is equivalent to assuming f 1 -alternating, left continuous, and f(0) = 0. Moreover, for any 0 ≤ x < z in R D f = −μ([x, z[) by an application of the inclusion/exclusion principle. Now, if f is additionally assumed to be homogeneous, that is f(tx) = tf(x) for any positive t and vector x then the measure μ is homogeneous: μ(tA) = tμ(A) for any positive t and measurable subset A (and reciprocally). Note that any homogeneous 1 -alternating function f : R ! R is automatically continuous, non-negative, with f(0) = 0. 1 This notion was rst introduced in [15] under the name fully d-max-decreasing. 182 Ë Cécile Mercadier and Paul Ressel An important example of a homogeneous measure is given by the image λ of the Lebesgue measure λ on R under the mapping s 7! s/w, where w 2 C. The co-survival function of λ is then + w λ ([x,∞] ) = λ(fs 2 R js/w ≱ xg) w + = λ(fs 2 R j9i ≤ d, s/w < x g) + i i = λ(fs 2 R js < max (x w )g) + i i i=1,...,d = max(x · w) . These functions will play a decisive role in the following, since they are the “building stones” of all homoge- neous co-survival functions. More precisely, consider the set of all normalized functions discussed above K := fψ : R ! Rjψ is 1 -alternating, homogeneous and ψ(1) = 1g . Then K is obviously convex and compact (with respect to pointwise convergence). It turns out that K is even a simplex, with fx 7! max(x · w)jw 2 Cg = ex(K) as its set of extreme points, and this set is closed (so compact as well) ; see [16, Theorem 4 (ii)]. In other words, K is a so-called Bauer simplex, i.e. for each ψ 2 K the representing probability measure on ex(K) guaranteed by Krein–Milman’s theorem, is unique. The resulting integral representation is also called Choquet represen- tation. So, for each 1 -alternating and homogeneous ψ on R , ψ ̸ 0, there is a unique probability measure ν on C such that Z ψ(x) = ψ(1) max(x · w)dν(w), x 2 R . It is easily seen that ψ is the co-survival function of the measure μ := ψ(1) λ dν(w). 2.2 Expression of Sobol eects and associated variances The main result of this paper is stated below. It says that Sobol eects ψ (as their variances) have rather simpler expressions in comparison with (2) when ψ is a homogeneous co-survival function. Indeed, they are expressed as integrals on C × [0, 1] of rank-one functions. Recall that 1 = (1, . . . , 1) in R . Theorem 1. Let ψ be a homogeneous co-survival function (3) associated with a spectral probability measure ν on C. Then, the term ψ in the Hoeding–Sobol decomposition with respect to λ satises on [0, 1] 8 9 Z Z < = Y Y ψ (x) = −ψ(1) 1 − K (w ; s) K (w ; s) ds dν(w) u s≥x w i i i i i i : ; C 0 i2u i2̸u for any non-empty subset u off1, . . . , dg and ( ) ! Z Z ψ = ψ(1) 1 − K (w ; s)ds dν(w) . i i C 0 i=1 Its corresponding variance ψ has the following expression Z Z Z Z 1 1 Y Y 2 2 σ = ψ(1) dν(w) dν(v) ds dt K (w ; s)K (v ; t) K (w , v ; s, t) − K (w ; s)K (v ; t) . u i i i i i i i i i i i C C 0 0 i2̸u i2u Furthermore, Z Z Z Z 1 1 2 2 2 σ = −(ψ(1) − ψ ) + ψ(1) dν(w) dν(v) ds dt K (w , v ; s, t) . ? i i i C C 0 0 i=1 Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 183 The link of Sobol eects ψ and their variances σ with the spectral measure ν has been made explicit. The main ingredient for proving the previous theorem is to remark that the spectral representation of ψ can be written as an integral of rank-one tensors. Then, all Sobol eects ψ and corresponding variances σ in- herit the same form by application of the Fubini–Tonelli theorem. As can be seen through Formula (2), the variance σ is usually computed as an alternating combination of cumulated variances. It thus suers from accumulation of estimation error, overall as d becomes larger. Theorem 1 oers a setting where the numerical 2 2 complexity of σ is the same as that of σ or other well-known quantities discussed in Subsection 2.3. Example 1. If the measure λ corresponds to the product of Lebesgue measures dλ(x) = dx ··· dx then 1 d K (w, v; s, t) = (s/w)^ (t/v)^ 1. Under this measure λ, consider ψ(x) = max(x), a particular extreme point. By 2 2 Theorem 1, with the probability measure ν = δ on C, one obtains ψ = d/(d + 1), σ = d/((d + 1) (d + 2)), 1 ? d−juj ψ (x) = − 1 − s s ds u ( s≥x ) i2u and Z Z 1 t 2(2d −juj + 1)!juj! 2 d−juj juj d σ = 2 t (1 − t) s ds dt = . (d + 1)(2d + 2)! 0 0 Example 2. Consider an extreme point of the convex and compact set K (mentioned in Subsection 2.1), precisely ψ(x) = max(x · w) with w 2 C. It is worth noticing that Theorem 1 furnishes the expressions of the variances σ 2 2 and σ as integrals on [0, 1] of a product of d univariate functions. In comparison with their original denitions, already mentioned in the introduction, this provides an important gain: The number of integrals is reduced (it is no longer an alternating sum) and the domain of integration is smaller. Under a precise value of w, the calculations would give exact expressions after very tedious eorts. One could numerically approximate them by Monte-Carlo procedures on [0, 1] instead. With λ as the two dimensional Lebesgue measure, we focus here on these extreme points in the bivariate setting. For w = (w, 1), we obtain 2 2 2 4 3 ψ = 1/2 + w /6 and σ = 1/12 − w /6 − w /36 + w /6 , with the following decomposition of σ 2 4 2 2 3 4 2 3 4 σ = w /45, σ = −w /6 + 2w /15 − w /36 + 1/12, σ = w /30 − w /45 . f1g f2g f1,2g 2.3 Consequences for cumulated variances It turns out that several combinations of variances are of prime interest in order to characterize the importance of a subset u of variables. Justications can be found in [9, 18] in the case of 2 2 u −u u −u 2 I = σ = ψ(x , x )ψ(x , z )dλ(x)dλ (z) − ψ (4) −u u v ? 2d−juj [0,1] vu and 2 2 2 2 u −u u −u τ = σ = σ + ψ − ψ(x , x )ψ(z , x )dλ(x)dλ (z) . (5) u v ? d+juj [0,1] v\u≠? 2 2 2 2 2 2 We see immediately that 0 ≤ I ≤ τ ≤ σ and I + τ = σ . Finally, [6] examined the meaning of the sum u u u −u over the supersets of u 2 2 = σ . (6) u v vu 184 Ë Cécile Mercadier and Paul Ressel 2 u Ranking based on the superset quantities takes into account the importance of x but additionnaly that of any vector containing these juj variables. Formulae depending on the spectral measure are now derived for these three types of cumulated variances. The next corollary asserts that they are also written as integrals of rank-one tensors. Corollary 1. Let ψ be a homogeneous co-survival function (3) associated with a spectral probability measure ν on C. Then, Z Z Z Z 1 1 Y Y 2 2 2 I = −(ψ(1) − ψ ) + ψ(1) dν(w) dν(v) ds dt K (w , v ; s, t) K (w ; s)K (v ; t) u ? i i i i i i i C C 0 0 i2u i2̸u Z Z Z Z 1 1 2 2 τ = ψ(1)(2ψ − ψ(1)) + ψ(1) dν(w) dν(v) ds dt u ? C C 0 0 Y Y Y K (w , v ; s, t) − K (w ; s)K (v ; t) K (w , v ; s, t) i i i i i i i i i i i2u i2u i2̸u Z Z Z Z 1 1 Y Y 2 2 = ψ(1) dν(w) dν(v) ds dt K (w , v ; s, t) K (w , v ; s, t) − K (w ; s)K (v ; t) u i i i i i i i i i i C C 0 0 i2̸u i2u The Choquet representation of will play a crucial role in the proof of the upper bound stated in the extreme value theory setting at the end of the paper. Example 3. Consider again Example 1. One obtains easily Z Z 1 t juj 2 d−juj d 2 I = 2 t s ds dt − (1 − ψ ) = , u ? (d + 1) (2d −juj + 2) 0 0 Z Z 1 t d 2 2 2 2 juj d τ = σ + ψ − 2 t s ds dt = − , u ? d + 2 (d + 1)(d +juj + 2) 0 0 and Z Z 1 t 2 d!juj! 2 juj d = 2 (1 − t) s ds dt = . (d +juj + 2)! 0 0 In the opinion of the authors the current example (as its rst part Example 1) looks promising for being a con- venient test function. It provides a simple but non trivial function which has known individual variances as well as cumulated and global ones, for any dimension d. In [9, Theorem 1] the following identity is shown 2 −juj x −u 2 u = 2 (D u ψ(·, z )) dx dz (7) u z d+juj [0,1] −u u −u u u −u 2 where ψ(·, z ) : [0, 1] ! R is dened by ψ(·, z )(x ) := ψ(x , z ). The gain of the expression of claimed in Corollary 1 can be questioned with regard to the dimension of the domain of integration. Similar 2 2 2 comments hold for I and τ with reference formulae (4) and (5). But, it does not exist a direct formula of σ , u u u i.e. based on ψ, except from inversion of (6) for instance. It yields X X 2 jv\uj 2 jv\uj −jvj x −v 2 v σ = (−1) = (−1) 2 (D v ψ(·, z )) dx dz . (8) u v z d+jvj [0,1] vu vu Comparing the already known formula (8) with our result associated with σ in Theorem 1 makes the interest of our expressions more obvious. In Theorem 1 indeed, it is no longer expressed as an alternating sum of integrals. We have reduced the dimension of integration. Nevertheless, to be also numerically convincing, a wide comparison between the estimation of σ derived from (8) and from Theorem 1 will now be oered. The same comparison is done for the estimation of , Formula (6) competing with the one from Corollary 1. u Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 185 2.4 Numerical illustrations For the sake of simplicity, we assume here that the distribution of the entries are known and xed as uni- forms. Our goal is to compare the eectiveness of the new formulae, obtained for homogeneous co-survival functions, with the already known and general ones. Both are integrations approximated by Monte-Carlo procedures, but neither the domain of integration nor the complexity of the integrand are the same. Our choices must assess impartiality. One possibility is to compare the estimation obtained after a given common executing time. However, this depends strongly on the way the integrands are coded. We thus decide to x the Monte-Carlo size N on the unit interval. We will rst restrict ourselves to the case of the max function ψ(x) = max(x) for which the exact values are known (see Examples 1 and 3). This corresponds to a one-point measure. The comparison will therefore be broken down according to the value of d and the size of u with respect to d. Completely arbitrarily we set the following values d = 5 or d = 10 ; then u = f1, 2g or u = f1, . . . , dg. Both 2 2 σ and will be estimated. The measures are based on the absolute mean error obtained over n replicates u u and dened as AME := jθ − θ j i,N 0 i=1 where θ is the true value and θ is the i-th estimate. The number of replicates here is n = 50. i,N Table 1: AME for the estimation of σ when ψ(x) = max(x). Missing value – refers to exceeding the time limit. d = 5 d = 10 u = f1, 2g u = f1, . . . , dg u = f1, 2g u = f1, . . . , dg −5 −5 −5 −9 Formula (8) N = 100 48.25 × 10 2.6 × 10 17.52 × 10 24.86 × 10 −5 −5 −5 −9 Theorem 1 N = 100 8.34 × 10 1.22 × 10 1.19 × 10 6.22 × 10 Formula (8) N = 10, 000 − − − − −5 −5 −5 −9 Theorem 1 N = 10, 000 0.86 × 10 0.11 × 10 0.1 × 10 0.75 × 10 The level of accuracy is the same on each column of Table 1 in order to facilitate the comparison. Two values for N have been handled: N = 100 and N = 10, 000. However, for the largest value, the time limit has been reached using the already known formula. Table 2: AME for the estimation of when ψ(x) = max(x). Missing value – refers to exceeding the time limit. d = 5 d = 10 u = f1, 2g u = f1, . . . , dg u = f1, 2g u = f1, . . . , dg −5 −5 −6 −10 Formula (7) N = 1000 12.15 × 10 79.41 × 10 31.18 × 10 159 × 10 −5 −5 −6 −10 Corollary 1 N = 1000 7.15 × 10 0.40 × 10 14.71 × 10 24.39 × 10 −5 −5 −6 Formula (7) N = 10, 000 5.43 × 10 0.26 × 10 12.07 × 10 − −5 −5 −6 −10 Corollary 1 N = 10, 000 1.99 × 10 0.11 × 10 3.68 × 10 7.53 × 10 In Table 2 only one estimation has not been obtained because of time exceedance. Again, the level of accuracy is the same on each column to facilitate the comparison. 186 Ë Cécile Mercadier and Paul Ressel Let us now consider another homogeneous co-survival function associated with a discrete probability measure ν = p δ where each w lies in C and p + . . . + p = 1 so that w m k k 1 k=1 k ψ(x) = p max(x·w ) . (9) k k k=1 Fix arbitrarily m = 15 and d = 5. The weights, chosen at random, are (p , . . . , p ) = (0.04, 0.08, 0.12, 0.05, 0.02, 0.10, 0.11, 0.01, 0.12, 0.13, 0.06, 0.03, 0.10, 0.01, 0.02) and the associated locations (w , . . . , w ) are 1 m 0 1 0.11 1.00 0.52 0.21 0.38 1.00 1.00 0.36 1.00 0.18 0.18 0.20 0.17 0.02 0.31 B C 0.62 0.81 0.59 0.52 1.00 0.56 0.59 0.08 0.15 0.10 0.35 1.00 0.56 0.43 1.00 B C B C B 0.67 0.84 1.00 0.24 0.43 0.69 0.12 0.20 0.09 0.71 0.62 0.31 1.00 0.37 0.04 C . B C @ A 1.00 0.65 0.64 0.41 0.76 0.74 0.57 1.00 0.49 1.00 1.00 0.54 0.42 1.00 0.44 0.32 0.37 0.03 1.00 0.02 0.11 0.50 0.70 0.18 0.16 0.75 0.03 0.11 0.18 0.29 Since the true values are not easily computable, we only provide a graphical comparison of the resulting boxplots obtained from n = 50 repetitions. (a) (b) (a) (b) u = f1, 2g u = f1, . . . , dg Figure 1: Estimation of σ when ψ is given by (9) for u = f1, 2g on the left panel and u = f1, . . . , dg on the right. The boxplot (a) is associated to the well-known Formula (8) whereas (b) refers to the new one stated in Theorem 1. As expected, this numerical study shows that the estimation from the new formulae is more accurate. This is nothing more than the illustration of the domain of integration being reduced. The reader should be aware that recent studies in GSA provided new methods compared to the classical Monte-Carlo procedure. Going further with a comparison based on pick-freeze method or any other renement would clearly exceed our ambitions in this paper. 3 Statistical applications in extreme value theory In the following, we focus on the Hoeding–Sobol representation of a stable tail dependence function (stdf). A homogeneous co-survival function ` is a stdf i `(e ) = . . . = `(e ) = 1 i.e., it is associated with a probability 1 d 5e-05 6e-05 7e-05 8e-05 9e-05 4.0e-08 6.0e-08 8.0e-08 1.0e-07 1.2e-07 1.4e-07 Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 187 (a) (b) (a) (b) u = f1, 2g u = f1, . . . , dg Figure 2: Estimation of when ψ is given by (9) for u = f1, 2g on the left panel and u = f1, . . . , dg on the right. The boxplot (a) is associated to the well-known Formula (7) whereas (b) refers to the new one stated in Corollary 1. measure ν on C satisfying w dν(w) = 1/`(1) 8 i = 1, . . . , d . Denote by μ the measure such that `(x) = μ([x,∞] ). This measure μ is closely related to the so-called expo- nent measure μ introduced in [14, Section 5.4.1] for instance. In fact, for any x 2 R * + c c μ([x,∞] ) = μ ([0, 1/x] ) . This means that μ is the image of μ under x 7! 1/x, so that μ is directly homogeneous (as is `) when μ is * * −1 d inversely homogeneous: μ (tA) = t μ (A) for any positive t and any measurable set A of [0,∞] \f0g. * * Whereas the characterization of stdfs was shown relatively late [16, Theorem 6], their integral representation was known long before: it goes back essentially to [4]. Most of the use of their integral representation has been done under the L or L -norm on R . But as emphasized by de Haan and Resnick, it is an arbitrary 1 2 + choice. As seen in Section 2, the extreme points of K (functions x 7! max(x· w) for w 2 C) combined with the max-norm were natural choices here. The main objective of this section is to analyze the theoretical aspect of the functional decompositon for stdfs with respect to the Lebesgue measure dλ(x) = dx . . . dx . As mentioned in the introduction, this idea 1 d has been introduced in [10] but the focus was on the meaning of in this context, named as tail superset indices, and on their estimation. To illustrate their importance in multivariate extreme value modeling, let 2 2 us focus for instance on the comparison (`) < (`). This means that the asymptotic dependence fi,jg fh,kg between components X and X themselves added to the asymptotic dependence between the pair (X , X ) and i j i j the d − 2 remaining variables is weaker than its equivalent in h, k. Reducing the dimension of the asymptotic dependence structure consists in selecting subsets u according to their tail superset indices . Below, we rst obtain a simplied expression for these indices by application of Corollary 1 to `. Then, we deduce an upper bound for the tail superset indices. The section is ended by a short discussion. 3.1 Tail superset indices The tail dependograph introduced in [10] starts from a non-oriented graph whose vertices represent compo- nents of the random vector X in the domain of attraction of `. The edge between i and j is drawn proportionally to the pairwise superset indices of `. This index measures the strength of asymptotic dependence be- fi,jg tween the components X and X , not only in their associated bivariate model (X , X ), but in the complete i j i j 0.00009 0.00010 0.00011 0.00012 0.00013 0.00014 0.00015 4.0e-08 6.0e-08 8.0e-08 1.0e-07 1.2e-07 1.4e-07 1.6e-07 188 Ë Cécile Mercadier and Paul Ressel model X. A thick line reveals a strong asymptotic dependence between corresponding components, whereas at the opposite, such index vanishes when the asymptotic dependence is null. The present paper thus oers a theoretical expression of the tail dependograph indices as Z Z Z Z 1 1 s t 2 2 (`) = `(1) dν(w) dν(v) ds dt ^ ^ 1 fi,jg w v k k C C 0 0 k≠i,j s t s t s t s t ^ ^ 1 − ^ 1 · ^ 1 ^ ^ 1 − ^ 1 · ^ 1 . w v w v w v w v i i i i j j j j Pairwise indices are perhaps the most important since their value on a graph is easily represented by the thickness of a segment. However, more general indices can be dened and an application of the previous section also provides the representation of (`) as follows Z Z Z Z 1 1 Y Y s t s t s t `(1) dν(w) dν(v) ds dt ^ ^ 1 ^ ^ 1 − ^ 1 · ^ 1 . w v w v w v i i i i i i C C 0 0 i2̸u i2u + + Examples. The asymptotic independence occurs when ` (x) := x so that ` (1) = d and i=1 2 + ν = ( δ )/d. All the terms in the integrand of (` ) cancel since at least the term depending i=1 fi,jg 2 + on i or on j (or both) will be reduced to (1 − 1). As a consequence (` ) = 0 as soon asjuj ≥ 2. _ _ The asymptotic complete dependence corresponds to ` (x) := max(x) so that ` (1) = 1 and ν = δ . 2 _ All indices of interest (` ) are equal to fi,jg Z Z Z 1 1 1 2 4 2 _ d−2 2 2 d+1 (` ) = ds dt(s ^ t) s ^ t − s · t = (1 − t) t dt = . ( ) fi,jg d + 1 (d + 1)(d + 2)(d + 3)(d + 4) 0 0 0 For in between strengths of asymptotic dependence, one can use logistic extreme value models. Symmetric versions 1/r 1/r `(x) = x + . . . + x for r 2 (0, 1) are obtained with `(1) = d and ν(dw) = q (w )1 d−1dw i −i w =1,w 2[0,1] i −i i=1 where for instance ! ! r−d −(1/r+1) d−1 d−1 X Y −1/r q (w ) = c w + 1 w d −d i i=1 i=1 as pointed out in [16]. Unfortunately, the expressions obtained in the present paper do not allow a real sim- plication under such models. 3.2 Upper bounds for tail superset indices Some simple computations allow to obtain the following lower and upper bounds. Lemma 1. Let ` be a d-variate stable tail dependence function. Then, d d ≤ ` ≤ d + 1 2 and d d d 2 2 ≤ σ (`) + ` ≤ + , d + 2 12 4 Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 189 implying 3 2 d (3d + 7d − 7d + 1) σ (`) ≤ . 12(d + 1) _,u Moreover, for any subset u off1, . . . , dg, set ` (x) = max x . Then, i2u i d juj 2 _,u _,u 2 2 2 I (` ) + (` ) ≤ I (`) + ` ≤ + . u u ? 4 12 The lower bound is given by _,u _,u 2 _,u 2 σ (` ) + (` ) 2` (d −juj) (d −juj) 2 _,u _,u 2 ? ? I (` ) + (` ) = + + 2 2 (d −juj + 1) d −juj + 1 (d −juj + 1) where juj juj _,u 2 _,u ` = and σ (` ) = . juj + 1 (juj + 1) (juj + 2) Hence, one can derive lower and upper bounds for I of a stable tail dependence function in any dimension, and for any size of the subset u, whenever ` is controlled. However, it doesn’t provide a very simple way to 2 2 deduce bounds for σ or, more interestingly in the tail dependograph context, for . The following result u u answers this question. Theorem 2. Let ` be a d-variate stable tail dependence function. Then, for any non-empty u f1, . . . , dg 2(juj!) 2 2 _,u (`) ≤ (` ) = u u (2juj + 2)! _,u u where ` (x ) = max x . i2u i 2 2 _,u If ` is a d-variate stdf with equality (`) = (` ) for a given ? ≠ u f1, . . . , dg, then its projection on the u u u u −u _,u u variables x is equal to `(x , 0 ) = ` (x ) = max x . i2u i 2 2(d!) In particular, if ` is a d-variate stdf with equality (`) = then `(x) = max(x). f1,...,dg (2d+2)! The proof is postponed to Section 4. However, note that it relies on the following preliminary result. Proposition 1. Let f : R ! R be 1 -alternating and let u be a non-empty subset off1, . . . , dg. Then, 2 2 [u] (f) ≤ (f ) u u [u] u u −u with f (z ) := f(z , 0 ). The authors conjectured the sharp upper bound in Theorem 2 a long time ago but the rigorous proof was only made possible after transferring the Choquet representation of the function ` to its indices as investigated in Section 2. The optimization problem dealt with in Theorem 2 might be looked at in the broader perspective of maximizing a convex functional over a compact convex set (which need not be a simplex). Bauer’s maximum principle ensures that the maximal value is attained in an extreme point, in our case in an extreme stable tail dependence function. It does however give no hint to localize such a point nor to its uniqueness. Our statement in Theorem 2 answers completely the question: it asserts the existence, the uniqueness and the location (and so nds the maximal value) of the maximization problem. The following statement is included in the proof of Theorem 2. Corollary 2. Let ψ be a homogeneous co-survival function (3) associated with a spectral probability measure ν on C. Then, 2(d!) 2 2 1/d (ψ) ≤ ψ(1) w dν(w) . f1,...,dg i (2d + 2)! i=1 190 Ë Cécile Mercadier and Paul Ressel 3.3 Practical meaning and use Taking into account the bounds provided by Theorem 2, the tail superset indices are normalized after multi- plication by 90, so that the corresponding anity matrix is [90 (`)] , on which classical clustering 1≤i,j≤d fi,jg algorithms and analyses can be performed. However, even if this pairwise normalization is correct, the use of this bound is more powerful when com- 2 2 _,u paring subsets with dierent sizes. Indeed, thanks to the renormalization (`)/ (` ) due to Theorem 2 u u all the renormalized superset tail indices can now be compared even if the subsets u have unequal sizes. The eective dimension of the asymptotic dependence structure could be dened as (2juj + 2)! (`) Δ(`) := argmax . juj,uf1,...,dg 2(juj!) + 2 In the asymptotic independent case, Δ(` ) = 1 since always vanishes: the entire additive component fi,jg + _ of ` explains the whole variance. In the asymptotic complete dependent case, Δ(` ) = d by application of Theorem 2. Now, for models in between, rules can be easily dened: select a subset u that achieves the maximization, or remove (in the asymptotic dependence modeling) subsets associated with small values of the previous brace. Let us provide an example. Consider ` a trivariate stdf with value at (x, y, z) given by 1 1 1 1 1 1 0.8 0.44 0.67 0.8 0.8 0.44 0.44 0.67 0.67 `(x, y, z) = ((0.36x) + (0.35y) ) + ((0.37x) + (0.38z) ) + ((0.32y) + (0.30z) ) 1 1 1 0.04 0.04 0.04 0.04 + ((0.27x) + (0.33y) + (0.32z) ) . (10) It is an asymmetric extreme value logistic model. Its associated tail dependograph is drawn below Figure 3: The Tail Dependograph of the stdf (10). 2 −4 2 −4 2 −4 2 A quick estimation gives = 1.700684 × 10 , = 1.625145 × 10 , = 2.909913 × 10 and = 12 13 23 123 −4 1.391676×10 . These values are not completely comparable since the sizes of the subsets are unequal. Their single use does not indicate the eective dimension of the function `. Applying our bound, one obtains (`)· −2 _,12 2 −2 _,13 2 −2 _,23 2 −2 _,123 (` ) ' 0.0153, (`) · (` ) ' 0.0146, (`) · (` ) ' 0.0262 and (`) · (` ) ' 12 13 13 23 23 123 123 0.0779. These calculations reveal that the strength of asymptotic dependence, when modelled by `, between the three components is closer to the possible maximal value, than its pairwise equivalent. Thus, the eective dimension of ` is 3 and it is not 2, according to this criteria. In other words, one should not simplify the representation of ` by combining only bivariate terms. The knowledge of our bound is crucial to construct this reasoning. Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 191 4 Proofs Proof of Theorem 1. The proof relies on the combination of Z Z Z ψ(x) = ψ(1) max(x · w)dν(w) = ψ(1) − ψ(1) dν(w) ds1 s≥x·w C C 0 for x 2 [0, 1] with the equality ψ (x) = ψ(x)dλ (x) . v −u −u [0,1] vu Then, applying the Fubini–Tonelli theorem yields Z Z Z juj ψ (x) = 2 ψ(1) − ψ(1) dλ (x) dν(w) ds1 v −u s≥x·w −u [0,1] C 0 vu 8 9 Z Z < = Y Y juj = 2 ψ(1) − ψ(1) 1 K (w ; s) ds dν(w) . s≥x w i i i i : ; C 0 i2u i2̸u Consequently, if u = ? ( ) Z Z ψ = ψ(1) − ψ(1) K (w ; s) ds dν(w) ? i i C 0 i=1 whereas if u ≠ ? 8 9 < = X X ju\vj ψ (x) = (−1) ψ (x) ˜v : ; vu ˜vv 8 9 Z Z < = X Y Y ju\vj = −ψ(1) (−1) 1 K (w ; s) ds dν(w) s≥x w i i i i : ; C 0 vu i2v i2̸v 0 1 Z Z X Y Y Y ju\vj @ A = −ψ(1) (−1) 1 K (w ; s) K (w ; s) dsdν(w) s≥x w i i i i i i C 0 vu i2v i2̸u i2u\v 8 9 Z Z < 1 = Y Y = −ψ(1) 1 − K (w ; s) K (w ; s) ds dν(w) . s≥x w i i i i i i : ; C 0 i2u i2̸u For non-empty u, the term ψ is centered so that its variance σ is also the second order moment. Z Z Z Z Z 1 1 2 2 ψ (x )dλ (x) = ψ(1) dν(w) dν(v) ds dt K (w ; s)K (v ; t) u u u i i i i juj [0,1] C C 0 0 i2̸u 1 − K (w ; s) 1 − K (v ; t) dλ (x) s≥x w t≥x v u i i i i i i i i juj [0,1] i2u Z Z Z Z 1 1 = ψ(1) dν(w) dν(v) ds dt K (w ; s)K (v ; t) i i i i C C 0 0 i2̸u 1 − K (w ; s) 1 − K (v ; t) dλ (x ) s≥x w i i i i t≥x v i i i i i i i2u Z Z Z Z 1 1 = ψ(1) dν(w) dν(v) ds dt K (w ; s)K (v ; t) i i i i C C 0 0 i2̸u K (w , v ; s, t) − K (w ; s)K (v ; t) . i i i i i i i i2u 192 Ë Cécile Mercadier and Paul Ressel 2 2 2 The last assertion comes from the computation of ψ dλ(x) = σ + ψ . More precisely, Z Z Z Z Z 1 1 2 2 2 σ + ψ = ψ(1) dν(w) dν(v) ds dt (1 − 1 )(1 − 1 )dλ(x) s≥x·w t≥x·v C C 0 0 [0,1] Z Z Z Z 1 1 = ψ(1)(2ψ − ψ(1)) + ψ(1) dν(w) dν(v) ds dt K (w , v ; s, t) ? i i i C C 0 0 i=1 R R R 2 2 using the fact that ψ(1) 1 dν(w)dλ(x)ds = ψ(1) − ψ(1)ψ . The result follows. s≥x·w d ? C [0,1] [0,1] Proof of Corollary 1. Following [18] one knows that 2 2 u −u u −u ψ + I = ψ(x , x )ψ(x , z )dλ(x)dλ (z) ? u −u 2d−juj [0,1] Z Z Z Z Z 1 1 = ψ(1) dλ(x)dλ (z) dν(w) dν(v) ds dt −u 2d−juj [0,1] C C 0 0 u u −u −u u u −u −u (1 − 1 1 )(1 − 1 1 ) s≥x ·w s≥x ·w t≥x ·v t≥z ·v Z Z Z Z 1 1 = ψ(1)(2ψ − ψ(1)) + ψ(1) dν(w) dν(v) ds dt C C 0 0 dλ(x)dλ (z)1 u u1 −u −u1 u u1 −u −u −u s≥x ·w s≥x ·w t≥x ·v t≥z ·v 2d−juj [0,1] Z Z Z Z 1 1 = ψ(1)(2ψ − ψ(1)) + ψ(1) dν(w) dν(v) ds dt C C 0 0 Y Y K (w , v ; s, t) K (w ; s)K (v ; t) . i i i i i i i i2u i2̸u Again, starting from 2 2 2 u −u u −u σ + ψ − τ = ψ(x , x )ψ(z , x )dλ(x)dλ (z) ? u d+juj [0,1] Z Z Z Z Z 1 1 = ψ(1) dλ(x)dλ (z) dν(w) dν(v) ds dt 2d−juj [0,1] C C 0 0 u u −u −u u u −u −u (1 − 1 1 )(1 − 1 1 ) s≥x ·w s≥x ·w t≥z ·v t≥x ·v Z Z Z Z 1 1 = ψ(1)(2ψ − ψ(1)) + ψ(1) dν(w) dν(v) ds dt C C 0 0 Y Y K (w ; s)K (v ; t) K (w , v ; s, t) . i i i i i i i i2u i2̸u Now, applying (7), 0 1 2 ju\aj a −a @ A = (−1) ψ(x , z ) dλ (x)dλ(z) (11) juj 2 juj+d [0,1] au 0 0 0 ju\aj+ju\a j a −a a −a = (−1) ψ(x , z )ψ(x , z )dλ (x)dλ(z) juj 2 juj+d 0 [0,1] a,a u Z Z Z Z Z 1 1 = ψ(1) dν(w) dν(v) ds dt dλ (x)dλ(z) juj+d C C 0 0 [0,1] ju\aj+ju\a j a a −a −a 0 0 0 (−1) (1 − 1 1 )(1 − 1 1 ) s≥x ·w s≥z ·w a a 0 a −a − t≥x ·v juj t≥z ·v a,a u Z Z Z Z Z 1 1 = ψ(1) dν(w) dν(v) ds dt dλ (x)dλ(z) juj+d C C 0 0 [0,1] ju\aj+ju\a j a a −a −a 0 0 0 (−1) 1 1 1 1 s≥x ·w s≥z ·w a a 0 a −a − t≥x ·v juj t≥z ·v a,a u Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 193 ju\aj since (−1) = 0 as soon as u is non-empty. As usual, we let aΔb = (a[ b) \ (a\ b) be the symmetric au dierence of the subsets a and b. As a consequence, Z Z Z Z 1 1 2 2 = ψ(1) dν(w) dν(v) ds dt C C 0 0 X Y Y ju\aj+ju\a j (−1) K (w ; s)K (v ; t) K (w , v ; s, t) i i i i i i i juj 0 0 0 a,a u i2aΔa i2̸aΔa Z Z Z Z 1 1 X Y Y 2 jaj = ψ(1) dν(w) dν(v) ds dt (−1) K (w ; s)K (v ; t) K (w , v ; s, t) i i i i i i i C C 0 0 au i2a i2̸a where the last equality comes from the general fact X X ju\aj+ju\bj juj jaj (−1) f(aΔb) = 2 (−1) f(a) . au,bu au Finally, one obtains Z Z Z Z 1 1 Y Y 2 2 = ψ(1) dν(w) dν(v) ds dt K (w , v ; s, t) (K (w , v ; s, t) − K (w ; s)K (v ; t)) . i i i i i i i i i i C C 0 0 i2̸u i2u Proof of Proposition 1. Prior to proving Proposition 1, we review some preliminary results. Lemma 2. Let A , . . . , A be non-empty sets, A := A × . . . × A , and f : A ! R. For x and z both in A and for 1 d 1 d any subset v f1, . . . , dg then v −v (x ,z ) d−jvj x D f = (−1) D f . v −v z (z ,x ) In particular, z d x D f = (−1) D f . x z If x = z for some i 2 f1, . . . , dg then D f = 0. i i z 0 0 Proof of Lemma 2. For x and z both in A write x := (x , . . . , x ) and z := (z , . . . , z ). 1 1 d−1 d−1 Dene ρ : A ! R by ρ(t) := D 0 f(·, t) . Then x juj u −u D f = (−1) f(z , x ) uf1,...,dg X X juj u −u juj u −u = (−1) f(z , x ) + (−1) f(z , x ) uf1,...,d−1g vf1,...,d−1g,u=v[fdg X X juj u −(u[fdg) juj u −(u[fdg) = (−1) f(z , x , x ) − (−1) f(z , x , z ) d d uf1,...,d−1g uf1,...,d−1g = ρ(x ) − ρ(z ) d d = D ρ . It implies D f = 0 whenever x = z . Moreover, we also see that d d (x ,z ) x D f = −D f . 0 z (z ,x ) The analogue results hold for any coordinate i ≤ d. The conclusion follows by obvious iteration. Lemma 3. Let A , . . . , A R be non-empty, A := A × . . . × A , and suppose f : A ! R to be 1 -alternating. 1 d 1 d d d−1 0 0 Then, for any x and z both in A the function j=1 r(t) := D 0 f(·, t) is decreasing on A . d 194 Ë Cécile Mercadier and Paul Ressel 0 0 Proof of Lemma 3. By Lemma 2, we may assume that x ≤ z . For s ≤ t both in A we have 0 0 0 (x ,s) x x 0 ≥ D f = D 0 f(·, s) − D 0 f(·, t) 0 z z (z ,t) where both terms on the right hand side are non-positive. Hence 0 0 x x 0 ≥ D 0 f(·, t) ≥ D 0 f(·, s) z z and so r(t) ≤ r(s). We now go back to the proof of Proposition 1. By iterating Lemma 3, we deduce that X X u u ju\vj v −v x −u x −u ju\vj v u\v −u (−1) f(x , z ) = D u f(·, z ) ≤ D u f(·, 0 ) = (−1) f(x , z , 0 ) . z z vu vu It yields 0 1 2 ju\vj v u\v −u u u 2 [u] @ A (f) ≤ (−1) f(x , z , 0 ) dx dz = (f ) , u u juj vu [u] u u −u where f (z ) = f(z , 0 ). + _ d _,u Proof of Lemma 1. Recall that` (x) and` (x) now stand for x and max(x) respectively. Set also` (x) = i=1 max x . Stable tail dependence functions ` have the well-known property i2u i _ + ` ≤ ` ≤ ` . (12) To prove (12) recall the equality `(x) = μ([x,∞] ). Then, the inclusion c c [x e ,∞] [x,∞] = fyjy < x g i i i i i≤d leads to the result since ` is the identity on each axis. Indeed, ` is homogeneous and equals one at the canon- ical basis vectors. The inequality (12) is easily transferred to rst and second order moments _ + ` ≤ ` ≤ ` ? ? ? and 2 _ _ 2 2 2 2 + + 2 σ (` ) + (` ) ≤ σ (`) + ` ≤ σ (` ) + (` ) . ? ? ? Precisely, it yields d d ≤ ` ≤ d + 1 2 and d d d 2 2 ≤ σ (`) + ` ≤ + , d + 2 12 4 implying 3 2 d (3d + 7d − 7d + 1) σ (`) ≤ . 12(d + 1) From (12), one can also prove that d−juj+1 X X (max x ) + d −juj i2u i ≤ ` (x) = `(x)dx ≤ d/2 + fx − 1/2g . v −u d −juj + 1 vu i2u Indeed, the left hand term comes from d−juj−1 E[max(m, Y )] = max(m, y)(d −juj)y dy −u 0 Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 195 where m = max(x ) and Y is sampled from identical standard uniform. Taking moment of order 2, one u −u obtains 2 _,u _,u 2 2 2 2 + + 2 I (` ) + (` ) ≤ I (`) + ` ≤ I (` ) + (` ) . u u ? u ? Precise computations give d juj 2 + + 2 I (` ) + (` ) = + u ? 4 12 and 2 _,u _,u 2 _,u σ (` ) + (` ) 2` (d −juj) (d −juj) 2 _,u _,u 2 ? ? I (` ) + (` ) = + + 2 2 (d −juj + 1) d −juj + 1 (d −juj + 1) where juj _,u ` = juj + 1 and juj 2 _,u σ (` ) = (juj + 1) (juj + 2) are deduced from Example 1 (just replace d byjuj). Proof of Theorem 2. Proposition 1 allows us to focus on the majorization associated to the largest subset u in f1, . . . , dg. Then, for ` a d-variate stdf, we are just interested in an upper bound for . Let us introduce f1,...,dg the following notation h (x) = max(x · w) so that the spectral representation can be written as `(x) = `(1) h (x)dν(w) . From linearity, one obtains x x D ` = `(1) D h dν(w) . z z Now, recall μ is the image of the Lebesgue measure λ on the half-line R under the mapping s 7! s/w, and w + the fact that h (x) = μ ([x,∞] ) as discussed in Subsection 2.1. Consequently, w w jD h j = μ ([x^ z, x_ z]) = 1 ds . z w w x^z≤s/w≤x_z Then, from [9, Theorem 1] and what precedes, one can write Z Z 2 −d x 2 (`) = 2 (D `) dxdz f1,...,dg d d [0,1] [0,1] Z Z Z Z −d 2 x x ≤ 2 `(1) jD h jdν(w) jD h jdν(v) dxdz . z w z v d d [0,1] [0,1] C C The Fubini–Tonelli theorem leads to Z Z x x jD h jjD h jdxdz w v z z d d [0,1] [0,1] Z Z Z Z 1 1 = 1 1 dxdz dsdt x^z≤s/w≤x_z x^z≤t/v≤x_z d d 0 0 [0,1] [0,1] Z Z 1 1 s t s t = 2 ^ 1 − _ dsdt w v w v i i i i 0 0 i=1 1/d Z Z 1 1 d d s t s t ≤ 2 ^ 1 − _ dsdt w v w v i i i i 0 0 + i=1 1/d Z Z 1 1 d d d = 2 s ^ t 1 − s _ t w v dsdt ( ) ( ) i i 0 0 i=1 ! ! Z Z 1 1 d 1/d 1/d d d = 2 w v (s ^ t) (1 − s _ t) dsdt i i 0 0 i=1 196 Ë Cécile Mercadier and Paul Ressel where the inequality comes from the generalization of Hölder’s inequality: for any positive numbers p such 1 1 that + . . . + = 1 p p 1 n Z Z Z 1/p 1/p 1 n p p 1 n jf . . . f j dμ ≤ jf j dμ ··· jf j dμ . (13) 1 n 1 n Combining the last results gives us !2 ! Z Z Z 1 1 2 2 1/d d d (`) ≤ `(1) w dν(w) (s ^ t) (1 − s _ t) dsdt f1,...,dg C 0 0 i=1 ! ! 2/d Z Z Z 1 1 2 d d ≤ `(1) w dν(w) s ^ t 1 − s _ t dsdt ( ) ( ) C 0 0 i=1 invoking again (13). Since a stable tail dependence function satises `(e ) = 1, we have w dν(w) = 1/`(1). i i Finally Z Z 1 1 2(d!) 2 d d 2 (`) ≤ (s ^ t) (1 − s _ t) dsdt = (h ) = f1,...,dg f1,...,dg (2d + 2)! 0 0 by application of Corollary 1 with ψ(1) = 1, ν = δ , and Example 3. [u] Assume now the second assertion of Theorem 2 and recall that if ` is a d-variate stdf then ` is ajuj-variate stdf. Combining the assumption with what precedes and Proposition 1, we obtain 2 _,u 2 2 [u] 2 _,u (` ) = (`) ≤ (` ) ≤ (` ) u u u u 2 [u] _,u so that (` ) = (` ). The problem is now the same as the last statement of Theorem 2 for d = juj. The result will thus follow if one can prove it directly. Let ` be any stdf where the maximal value is attained. Then the inequalities after the inequality (13) are in fact equalities, in particular !1/d Z Z d d Y Y 1/d w dν(w) = w dν(w) C C i=1 i=1 1/d implying by Lemma 4 (below) that the functions w 7! w , 1 ≤ i ≤ d, are proportional, as are then also w 7! w . Since w dν(w) = 1/`(1) for each i, we see that ν-almost surely the components w are equal, i.e. i i i ν is concentrated on the diagonal fwjw = w = . . . = w g, and the only w 2 C with this property is w = 1. 1 2 d Consequently, ν = δ and `(1) = 1. In other words, `(x) = max(x , . . . , x ). 1 1 d Lemma 4. Consider in the generalized Hölder inequality (13) the special case p = . . . = p = 1/n, f ≥ 0 and 1 i 0 < f dμ < ∞ for all i. Then 1/n Z Z f . . . f dμ ≤ f dμ (14) 1 n i=1 with equality if and only if the functions f are proportional, i.e. f = α f for i = 2, . . . , n where α > 0. i i i 1 i R Q n n Proof of Lemma 4. If f = α f for all i (α = 1) then both sides in (14) have the same value f dμ · α . i i 1 1 i 1 i=1 Supposing now equality in (14), we proceed by induction. For n = 2 the inequality (14) is the Cauchy-Schwarz inequality and it is well-known that f and f are proportional in case of equality. We assume now the validity 1 2 of our assertion for some n ≥ 2 and consider n+1 functions f , . . . , f . Hölder’s inequality for two functions 1 n+1 g, h ≥ 0 reads Z Z Z 1/p 1/q p q ghdμ ≤ g dμ h dμ Hoeding–Sobol decomposition of homogeneous co-survival functions Ë 197 p q with p, q ≥ 1 such that 1/p + 1/q = 1, and with equality i g and h are proportional. It will be applied to g := f ··· f , h := f , p = (n + 1)/n and q = n + 1 1 n+1 n 1 Z Z Z n+1 n+1 n+1 n+1 (f ··· f )f dμ ≤ (f ··· f ) dμ f dμ n n 1 n+1 1 n+1 n n ! ! 1 1 Z n+1 Z Z n+1 n n+1 Y n n+1 Y n+1 n+1 n+1 ≤ f dμ f dμ = f dμ n+1 i i i=1 i=1 (n+1)/n (n+1)/n where the second majorization is obtained by applying the induction hypotheses to f , . . . , f . These two inequalities are by assumption equalities, and lead to n+1 n+1 f ··· f = β f ( n) 1 1 n+1 n+1 n+1 n+1 n+1 as well as f = β f , . . . , f = β f for positive β , β , . . . , β , i.e. n n 2 2 1 n 1 1 2 1/(n+1) 1/(n+1) β f = (β ··· β ) · f , n+1 2 n 1 showing f , . . . , f to be proportional. 1 n+1 Concluding remarks The Choquet representation of homogeneous co-survival functions, shown to be unique in [16], is the source of all results in this paper. Then, the Fubini–Tonelli theorem appears as the main ingredient in transposing the spectral expressions to several forms of combined variances. Illustrated through Monte-Carlo comparisons, the coverage accuracy is signicantly improved. However, this just illustrates the contraction of the domain of integration. As a natural example, the function that summarizes the tail dependence structure in extreme value theory, namely the stable tail dependence function, received successfully the application of the new results. Furthermore, the generalization of Hölder’s inequality associated with more tricky justications from multivariate monotonicity yielded a sharp upper bound to the quantity of interest. Finally, it may be worth pursuing the consequences for measuring the eective asymptotic dependence dimension of a random vector. Acknowledgements: We thank three referees for their comments and suggestions, which helped to improve our manuscript, and the editorial board for managing the overall review. The rst author thanks her colleague Guillaume Aubrun for generously participating in some discussions on convexity. 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Dependence Modeling – de Gruyter
Published: Jan 1, 2021
Keywords: Hoeffding–Sobol decomposition; co-survival function; spectral representation; stable tail dependence function; multivariate extreme value modeling; 26A48; 26B99; 44A30; 62G32; 62H05
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