Domination of Sample Maxima and Related Extremal Dependence Measures

For a given $d$-dimensional distribution function (df) $H$ we introduce the class of dependence measures $ \mu(H,Q) = - \mathbb{E}\{ \ln H(Z_1, \ldots, Z_d)\},$ where the random vector $(Z_1, \ldots, Z_d)$ has df $Q$ which has the same marginal df's as $H$. If both $H$ and $Q$ are max-stable df's, we show that for a df $F$ in the max-domain of attraction of $H$, this dependence measure explains the extremal dependence exhibited by $F$. Moreover we prove that $\mu(H,Q)$ is the limit of the probability that the maxima of a random sample from $F$ is marginally dominated by some random vector with df in the max-domain of attraction of $Q$. We show a similar result for the complete domination of the sample maxima which leads to another measure of dependence denoted by $\lambda(Q,H)$. In the literature $\lambda(H,H)$ with $H$ a max-stable df has been studied in the context of records, multiple maxima, concomitants of order statistics and concurrence probabilities. It turns out that both $\mu(H,Q)$ and $\lambda(Q,H)$ are closely related. If $H$ is max-stable we derive useful representations for both $\mu(H,Q)$ and $\lambda(Q,H)$. Our applications include equivalent conditions for $H$ to be a product df and $F$ to have asymptotically independent components.


Introduction
Let H be a d-dimensional distribution function (df) with unit Fréchet marginal df's Φ(x) = e −1/x , x > 0. We shall assume in the sequel that H is a max-stable df, which in our setup is equivalent with the homogeneity property H t (x 1 , . . . , x d ) = H(tx 1 , . . . , tx d ) (1.1) for any t > 0, x i ∈ (0, ∞), 1 ≤ i ≤ d, see e.g., [1][2][3]. The class of max-stable df's is very large with two extreme instances H 0 (x 1 , . . . , the product df H 0 and the upper df H ∞ , respectively. HereafterḠ = 1 − G stands for the survival function of some univariate df G. It follows easily by the lower Fréchet -Hoeding bound that (H(nx 1 , . . . , nx d )) n ≥ max 0, 1 − Indeed, (1.2) is well-known and follows for instance using the Pickands representation of H, see e.g., [3][Eq. From multivariate extreme value theory, see e.g., [1][2][3][4] we know that d-dimensional max-stable df's H are limiting df's of the component-wise maxima of d-dimensional independent and identically distributed (iid) random vectors with some df F . In that case, F is said to be in the max-domain of attraction (MDA) of H (abbreviated F ∈ M DA(H)). For simplicity we shall assume throughout in the following that F is a df on [0, ∞) d with marginal df's F i ∈ M DA(Φ), i ≤ d that have norming constants a n = n, n ∈ N, and thus we have In the special case that F has asymptotically independent marginal df's, meaning that for (X 1 , . . . , In various applications it is important to be able to determine if some max-stable df H resulting from the approximation in (1.5) is equal to H 0 , which in the light of multivariate extreme value theory means that the component-wise maxima M n := (max 1≤i≤n X i1 , . . . , max 1≤i≤n X id ), n ≥ 1 of a d-dimensional random sample (X i1 , . . . , X id ), i = 1, . . . , n of size n from F has asymptotically independent components.
The strength of dependence of the components of M n , or in other words the extremal dependence manifested in F , in view of the approximation (1.5) can be measured by calculating some appropriate dependence measure for H (when the limiting df H is known).
For any random vector Z = (Z 1 , . . . , Z d ) with df Q which has the same marginal df's as H we introduce a class of dependence measure for H indexed by Q given by In view of (1.3), since − ln H i (Z i ) is a unit exponential random variable, we have Next, consider the case that both H and Q are max-stable. It follows that (see Theorem 2.3) for F satisfying (1.5) and G ∈ M DA(Q) (1.10) provided that both F and G are continuous. In view of (1.10), we see that µ(H, Q) relates to F under (1.5).
Let in the following W denote a random vector with df G being independent of M n . We say that W marginally dominates M n , if there exists some i ≤ d such that W i > M ni . Consequently, assuming further that W is independent of M n we have Re-writing (1.10) we have lim n→∞ nπ n = µ(H, Q) and thus µ(H, Q) appears naturally in the context of marginal dominance of sample maxima.
Our motivation for introducing µ(H, Q) comes from results and ideas of A. Gnedin, see [5][6][7] where multiple maxima of random samples is investigated. In the turn, the probability of observing a multiple maximum is closely related to the complete domination of sample maxima as we shall explain below.
We say that W completely dominates M n if W i > M ni for any i ≤ d. Assuming that F and G are continuous, we (1.11) where υ denotes the exponent measure of H defined on E = [0, ∞] d \ (0, . . . , 0), see [1,3] for more details on the exponent measure. Note in passing that by symmetry lim n→∞ λ n (F n , G) = λ(H, Q) follows.
Our notation and definitions of π n and π n agree with those in [8] for the particular case that F = G. Therein the complete and simple records are discussed. If F is continuous and F = G we have that (n + 1)π n equals P max 1≤i≤n+1 X ij = X 1j , j = 1, . . . , d , which is the probability of observing a multiple maximum, see [6,7,[9][10][11][12]. There are only few contributions that discuss the asymptotics of λ n (G n , F ) for F = G, see [13][14][15].
Since the exponent measure can be defined also for max-id. df H, i.e., if H t is a df for any t > 0, then as above Brief organisation of the rest of the paper: In Section 2 we derive the basic properties of both measures of µ(H, Q) and λ(Q, H) if H is a max-id. df. More tractable formulas are then derived for H being a max-stable df. Section 3 is dedicated to applications. We present some auxiliary results in Section 4 followed by the proofs of the main results in Section 5.

Main Results
In the following H and Q are d-dimensional df's with unit Fréchet marginals df's and Z is a random vector with df Q. The second dependence measure λ(Q, H) defined in (1.11) is determined in terms of the exponent measure ν of H, under the max-stability assumption on H.
A larger class of multivariate df's is that of max-id. df's. Recall that H is max-id., if H t is a df for an t > 0. For such df's the corresponding exponent measure can be constructed, see e.g., [1], and therefore we can define λ(Q, H) as in the Introduction for any H a max-id. df and any given df Q. Note that any max-stable df is a max-id. df, therefore in the following we shall consider first the general case that H is a max-id. df, and then focus on the more tractable case that H is a max-stable df. However as we show below it is possible to calculate µ(H, Q) if we know λ(Q K , H K ) for any non-empty index set A similar result is shown for λ(Q, H). In our notation Q K denotes the marginal df of Q with respect to K and |K| stands for the number of the elements of the index set K. Below µ n and λ n are as defined in the Introduction.
ii) A direct consequence of (2.3) is that we can define λ(Q, H) even if H is not a max-id. df by simply using the iii) It is clear that µ(H, Q) ≥ µ(H K , Q K ) for any non-empty index set K ⊂ {1, . . . , d}. Note that (2.1) shows that exactly the opposite relation holds for λ(Q, H) when H is a max-id. df, namely In fact, (2.3) shows that we can calculate both µ(H, Q) and λ(Q, H) by a limit procedure if we assume that H is a max-id. df, see for more details (5.1). Although such a limit procedure shows how to interpret these dependence measures in terms of domination of random vectors, it does not give a precise relation with extremal properties of random samples. Therefore in the following we shall restrict our attention to the tractable case that H is a max-stable df.  Remark 2.4. The relation lim n→∞ λ n (F n , F ) = λ(H, H) for F ∈ M DA(H) is known from works of A. Gnedin, see e.g., [6,7]. Explicit formulas are given in [16] for d = 2. See also the recent contributions [8,12].
In view of [4] (recall H has unit Fréchet marginal df's) the assumption that H is max-stable implies the following de Haan representation (see e.g., [17,18]) As shown in [19], see also [20,21] we have the alternative Moreover, Ψ i 's are bounded by 1, which immediately implies the validity of the lower bound in (1.2).
Our next result gives alternative formulas for µ(H, Q) and shows that it is the extremal coefficient of the max-stable df H * defined by and Y i /Z i 's are non-negative, then H * has unit Fréchet marginal df's and moreover alsoH defined by is a max-stable df with unit Fréchet marginal df's.
Theorem 2.5. If H is a max-stable df with unit Fréchet marginal df 's and de Haan representation (2.6) with Y being independent of Z with df Q which has unit Fréchet marginal df 's, then we have Remark 2.6. i) If Z 1 = · · · = Z d = Z with Z a unit Fréchet random variable, then the zero-homogeneity of Ψ i 's, (2.5) and (2.8) imply that ii) In view of [12][Theorem 2.2] (see also [16][Eq. (6.9)]) for H with de Haan representation (2.6) holds, which together with (2.10) implies that and thus the lower bound in (1.8) is sharp. We note in passing that there are numerous papers where λ n (F n , F ) and λ(H, H) appear, see e.g., [8,16,[22][23][24] and the references therein.
iii) For common max-stable df 's H the spectral random vector Y that defines (2.4) is explicitly known. Consequently, for any given random vector Z, using the first expression in (2.8) and (2.9), we can easily evaluate µ(H, Q) and λ(Q, H) by Monte Carlo simulations, respectively.

Applications
In for any index set K ⊂ {1, . . . , d} with two elements. Therefore, in the sequel we consider for simplicity the case d = 2 discussing some tractable conditions that are equivalent with H = H 0 and (1.6).

Remark 3.4. i) The equivalence of i) and ii) in Proposition 3.3 is well-known and relates to Takahashi theorem,
i.e., it is enough to know that the limiting max-stable df H is a product df at one point, say (1,1 ii) Recall that the assumption F i ∈ M DA(Φ) means that lim n→∞ F n i (a ni x) = Φ(x), x ∈ R for some norming constants a ni > 0, n ∈ N. For notational simplicity, in this paper we assume that a ni 's equal n. If this is not the case, then we need to re-formulate statement ii) in Proposition 3.3 as n lim n→∞ nP{X 1 > a n1 , X 2 > a n2 } = 0. Note that if F ∈ M DA(H) with H a max-stable df, then In the literature, λ F is commonly referred to as the coefficient of upper tail dependence of F , see [3] for more details.

Auxiliary Results
if either of the limits exists. Further if if H denotes the df of (V 1 , . . . , V d ), then we have Proof of Proposition 4.2 For notational simplicity we consider below only the case d = 2. From the assumptions Let υ be the exponent measure of H defined on E, see [1] for details. For any x 0 , y 0 positive, since by our assumptions holds locally uniformly for (x 1 , x 2 ) ∈ (0, ∞) 2 , using further (4.8) and [19] Moreover, by (4.4) where the equality above is a consequence of the assumption that F n , G n have continuous marginal df's. Hence (4.5) follows and we show next (4.7). Similarly, for x 0 , y 0 as above lim sup hence the proof follows.
Remark 4.3. The validity of (4.4) has been shown under the assumption that G n is a continuous df. From the proof above it is easy to see that (4.4) still holds if we assume instead that G n is continuous and positive such that G n n is a df. Similarly, for the validity of (4.7) it is enough to assume that F n n is a continuous df.
By the assumptions since the df H is continuous, applying Theorem 2.3 and (4.1) with u, t > 0 we obtain establishing the proof.

Proofs
Proof of Theorem 2.1 For n > 0 set A n = Q 1/n and B n = H 1/n . Since H is a max-id. df, then B n is a df for The second claim in (2.1) follows with similar arguments and therefore we omit its proof.
Next, for any non-empty subset K of {1, . . . , d} with m = |K| elements by (2.1) where F nK , Q K are the marginals of F n and Q with respect to K. Note that for notational simplicity we write the marginal df's with respect to K as functions of x 1 , . . . , x d and not as functions of x j1 , . . . , x jm where K = {j 1 , . . . , j m } has m = |K| elements. By Fubini Theorem where F nK stands for the joint survival function of F nK . In the light of the inclusion-exclusion formula Using further the fact that H and Q have the same marginal df's, for any index set K with only one element we is valid for any i ≤ With the same arguments using now that E{1/Z i } = 1, i ≤ d we have The lower bound in (2.11) follows with similar arguments, hence the proof is complete.
if and only if lim n→∞ nP{G i (X i ) > 1 − 1/n} = c i or equivalently By the assumption c i ∈ (0, ∞) for i = 1, 2. Consequently, for all x > 0 there exist a 1 , a 2 positive such that Assume for simplicity that c i = 1, i = 1, 2. By the assumptions As in the proof of Proposition 4.2, using that F and G are in the MDA of H and Q, respectively, it follows that for and further establishing the proof.
Next, assume that ii) holds. We have that G(x 1 , x 2 ) ≤ G 1 (x 1 )G 2 (x 2 ) =: K(x 1 , x 2 ) and by the assumption that G i 's are in the MDA of Φ it follows that K is in the MDA of H ∞ . Further ii) implies that F ∈ M DA(H 0 ) and − ln H(1, 1) = 2. Consequently, Theorem 2.3 yields lim n→∞ λ n (K n , F ) = λ(H ∞ , H 0 ).
Let G be the joint survival function of the bivariate df G. For any positive integer n, we have that F n is a bivariate df. Hence by Fubini theorem and the fact that F i = G i , i = 1, 2 are continuous df's, for any positive integer n we obtain R 2 and thus the equivalence of iii) and iv) follows. The equivalence iv) and v) follows from Lemma 4.1 and thus the proof is complete.