M Ahmed

V Maume-Deschamps

P Ribereau

C Vial

SPATIAL RISK MEASURES FOR MAX-STABLE AND MAX-MIXTURE PROCESSES

Keywords: Risk measures, Spatial dependence, Max-stable process, Max-Mixture process, Extreme value theory. 1

. We evaluated the proposed risk measure by a simulation study.

Introduction

Storms are the most destructive natural hazards in Europe. The economic and the private sectors losses due to these extreme events are often important. For example, during December 1999, three storms hit Europe causing insured losses above 10 billion e (see [START_REF] Donat | High-resolution refinement of a storm loss model and estimation of return periods of loss-intensive storms over germany[END_REF][START_REF] Re | Winterstorms in europe ii-analysis of 1999 losses and loss potentials[END_REF][START_REF] Ulbrich | Three extreme storms over europe in december 1999[END_REF]). The storms may have a huge spatial component; in other words, the underlying spatial process may have a strong spatial dependence even at a long distance.

One of the main characteristics of climate events is the spatial dependence. Many dependence structures may arise: Asymptotic dependence; Asymptotic independence or both [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF]. The high impact of storm losses motivated us to propose risk measures taking into account the spatial dependence.

In case of univariate random variables, risk measures has been widely studied in the literature and the corresponding axiomatic formulation has been presented in [START_REF] Artzner | Coherent measures of risk[END_REF]. In [START_REF] Föllmer | Spatial risk measures and their local specification: The locally law-invariant case[END_REF] a collection of risk measures indexed by a network is introduced for some financial products. In spatial contexts, the spatial dependence plays an important role. For example, wind speed and rainfall amount e.g. have different spatial behavior, so that, after normalization of their marginal distributions, the value of a risk measure should not be the same. In [START_REF] Keef | Spatial risk assessment for extreme river flows[END_REF], the authors proposed to evaluate the risk on a region by a probability P(S > s) where S is an integrated damage function. In [START_REF]Spatial risk measures and applications to max-stable processes[END_REF] or [START_REF] Koch | Tools and models for the study of some spatial and network risks:application to climate extremes and contagion in france[END_REF] this idea is developed to define spatial risk measures taking into account the spatial dependence. In [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] the same idea is used: a risk measure constructed with the damage function D X,u = (X -u) + with u a fixed threshold for a Gaussian process X is studied. In the same spirit as [START_REF] Artzner | Coherent measures of risk[END_REF], the authors propose a set of axioms that a risk measure in the spatial context should verify. This point of view has been previously adopted in [START_REF]Spatial risk measures and applications to max-stable processes[END_REF] for some max-stable processes. Our main contributions concern the risk measure based on the intensity damage function D ν X = |X| ν with 0 < ν < 1/2, it consists in the development of the results from [START_REF]Spatial risk measures and applications to max-stable processes[END_REF]: further max-stable processes are involved and the computation technics are extended to max-mixture processes. We study the properties of the risk measure with respect to the parameters of each model (with a focus on the dependence parameter). We also study its axiomatic properties. This paper is organized as follows. Section 2 recalls definitions and properties of max-stable and max-mixture processes. In Section 3 we consider spatial risk measures and recall the axiomatic setting from [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] which derives from [START_REF]Spatial risk measures and applications to max-stable processes[END_REF]. Section 4 is devoted to the study of the risk measures with damage function |X| ν for max-stable and max-mixture processes. We propose forms of this risk measures and derive its behavior. We present in Section 5 a simulation study in order to evaluate this spatial risk measures. Concluding remarks are discussed in Section 6.

Spatial extreme processes

We shall focus on max-stable processes, inverse max-stable processes and maxmixtures of both, and we call these processes extreme processes. We shall emphasize on the modelization of the dependence structure and thus assume that the marginal laws have been normalized to unit Fréchet with distribution function F (x) = exp(-1/x), x > 0 (see [START_REF] Naveau | Modelling pairwise dependence of maxima in space[END_REF]).

2.1. Max-stable model. This is an extension of the multivariate extreme value theory to the spatial setting. We refer to [START_REF] Haan | A spectral representation for max-stable processes[END_REF][START_REF] De Haan | Extreme value theory: an introduction[END_REF] for definitions and properties of max-stable processes. We shall consider max-stable processes on S ⊂ R 2 with unit Fréchet marginal distributions (i.e. simple max-stable processes), for any (s, t) ∈ S 2 ,

P X(s) ≤ x 1 , X(t) ≤ x 2 =G s,t (x 1 , x 2 ) = exp(-V s,t (x 1 , x 2 )), (2.1)
where V s,t is the so-called exponent measure function. It is homogenous of order -1 and satisfies the bounds

(2.2) max{1/x 1 , 1/x 2 } ≤ V s,t (x 1 , , x 2 ) ≤ {1/x 1 + 1/x 2 }.
These bounds imply that X is positive quadrant dependent (PQD), see [START_REF] Lehmann | Some concepts of dependence[END_REF] for definitions and properties of PQD processes. In this paper, we consider stationary and isotropic processes. Thus, the exponent measure V s,t and the distribution function G s,t depend only on the norm h = ||s -t|| and will be denoted by V h and G h .

In [START_REF] Haan | A spectral representation for max-stable processes[END_REF], it is also proved that every simple max-stable process X has the following spectral representation:

X(s) = max i≥1 ξ i W i (s) s ∈ S,
where {ξ i , i ≥ 1} is an i.i.d Poisson point process on (0, ∞), with intensity dξ/ξ 2 and {W i , i ≥ 1} are i.i.d copies of a positive random field W = {W (s), s ∈ S}, such that E[W (s)] = 1 for all s and independent of ξ i .

Many dependence measures for spatial processes X have been introduced. These are generally bivariate dependence measures used in a spatial context. The tail dependence coefficient χ introduced in [START_REF] Ledford | Statistics for near independence in multivariate extreme values[END_REF] is defined by

(2.3) χ(h) = lim u→1 P F (X(s)) > u|F (X(s + h)) > u .
If χ(h) = 0, the pair (X(s + h), X(s)) is said to be asymptotically independent (AI). If χ(h) = 0, the pair (X(s + h), X(s)) is said to be asymptotically dependent (AD).

The process is said AI (resp. AD) if for all h ∈ S χ(h) = 0 (resp. χ(h) = 0). The extremal coefficient Θ(h) = V h (1, 1) satisfies χ(h) = 2 -Θ(h) and (see [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF])

(2.4) G h (x, x) = exp(-Θ(h)/x).
In [START_REF] Coles | Dependence measures for extreme value analyses[END_REF] an alternative definition of the tail dependence coefficient is given.

(2.5)

χ(h, u) = 2 - log P F (X(s)) < u, F (X(s + h)) < u log P F (X(s)) < u , 0 ≤ u ≤ 1.
We have lim u→1 χ(h, u) = χ(h).

The spectral representation is useful to construct specific max-stable processes. We present three of them: Smith, Schlater and truncated Schlater models.

Smith Model Introduced in [START_REF] Smith | Max-stable processes and spatial extremes[END_REF]. It is defined on S = R d . Its dependence structure is contained in a covariance matrix Σ. Let {(ξ i , s i )} be a Poisson point process on (0, ∞) × R d with intensity ξ -2 dξds and consider the d-dimensional Gaussian probability density function ϕ d (.; Σ) with mean 0 and covariance matrix Σ. For all s ∈ R d , define W i (s) = ϕ d (s -s i ; Σ) and

X(s) = max i≥1 {ξ i ϕ d (s -s i ; Σ)}.
The exponent measure function is given by

V h (x 1 , x 2 ) = 1 x 1 Φ τ (h) 2 + 1 τ (h) log x 2 x 1 + 1 x 2 Φ τ (h) 2 + 1 τ (h) log x 1 x 2 ; with τ (h) = √ h T Σ -1 h and Φ(•) the standard normal cumulative distribution func- tion.
The pairwise extremal coefficient equals

Θ(h) = 2Φ τ (h) 2 .
Note that if the covariance matrix is diagonal Σ = σI d , then the process X is isotropic as its bivariate distribution depends only on h through the function

τ (h) = 1 σ h .
Schlather Models This model introduced in [START_REF] Schlather | Models for stationary max-stable random fields[END_REF] provides a class based on a stationary Gaussian random field. Let W := {W (s), s ∈ S} be a stationary random field, with E W + (s) = µ ∈ (0, ∞) where W + (s) = max{0, W (s)}. Let {ξ i , i ≥ 1} be a Poisson point process on (0, ∞), with intensity dξ/ξ 2 and {W i , i ≥ 1} are iid copies of W (s). Consider

X(s) = µ -1 max i≥1 ξ i W + i (s), s ∈ S,
it defines a stationary max-stable process. Schlather proposed to take a stationary Gaussian process W (s) with correlation function ρ(•) and µ -1 = √ 2π. In this case, the resulting max-stable process X is called Extremal Gaussian process (EG). The exponent measure function is

V h (x 1 , x 2 ) = 1 2 1 x 1 + 1 x 2 1 + 1 -2(ρ(h) + 1) x 1 x 2 (x 1 + x 2 ) 2 .
The extremal coefficient is given by

Θ(h) = 1 + 1 -ρ(h) 2 1/2 .
We have lim h→∞ χ(h) = 0. In other words, the asymptotic dependence persists even at infinite distances. This might be unrealistic in applications. To overcome this problem a truncated version of W (s) can be used. Let {r i } be a homogenous

Poisson point process of unit rate on S and µ

-1 = √ 2π(E[|B|]) -1 .
Then, for a stationary Gaussian process W i (s), define (2.6)

X(s) = max i≥1 ξ i W i (s)1 Bi (s -r i ), s ∈ S
with B ⊂ S a compact random set and B i i.i.d. copies of B. The process X is a truncated extremal Gaussian process (TEG). The exponent measure function is given by

V h (x 1 , x 2 ) = 1 x 1 + 1 x 2 1 - α(h) 2 1 -1 -2(ρ(h) + 1) x 1 x 2 (x 1 + x 2 ) 2 .
The extremal coefficient is given by

Θ(h) = 2 -α(h) 1 - 1 -ρ(h) 2 1/2
where α(h)

= E{|B ∩ (h + B)|}/E[|B|].
Usually, B is a disk of radius r. In that case, α(h) = {1 -h/2r} + . For more details, see [START_REF] Davison | Geostatistics of extremes[END_REF]. That leads to χ(h) = 0, ∀h ≥ 2r. In other word the process X is asymptotically independent (and thus independent because it is max-stable) for all h ≥ 2r.

2.2.

Inverse max-stable processes. Max-stable processes are either AD or they are independent. This behavior may be unapropriate in applications: data may reveal asymptotic independence without being independent.

In [START_REF] Coles | Dependence measures for extreme value analyses[END_REF] the lower tail dependence coefficient χ(h) is proposed in order to study the strengt of dependence in AI cases.

(2.7)

χ(h) = lim u→1 2 log P F (X(s)) > u log P F (X(s)) > u, F (Y (s + h)) > u -1, 0 ≤ u ≤ 1.
We have -1 ≤ χ(h) ≤ 1 and the spatial process is asymptotically dependent if χ(h) = 1. Otherwise, it is asymptotically independent. We shall consider a class of asymptotically independent processes introduced in [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF]: Inverse max-stable process. Let X be a max-stable process with unit Fréchet margin, consider

X(s) = g(X(s)) = -1/ log{1 -e -1/X (s) } s ∈ S.
Then X is asymptotically independent with unit Fréchet margin and bivariate survivor function

P X(s 1 ) > x 1 , X(s + h) > x 2 = exp -V h g(x 1 ), g(x 2 ) .
where V h is the exponent measure function of X . We a slight langage abuse, we shall say that V h is the exponent measure function of X. Inverse max-stable processes enter in the class of processes defined in [START_REF] Ledford | Statistics for near independence in multivariate extreme values[END_REF] which satisfy:

P X(s) > x, X(s + h) > x = L h (x)x -1/η(h) , x → ∞
where L h (x) is a slowly varying function and η(h) ∈ (0, 1] is called the tail dependence coefficient. For these kind of processes, the AI is caracterized by η(h) < 1.

2.3.

Max-Mixture model. In spatial contexts, specifically in environmental domain many scenarios of dependence could arise and AD and AI might cohabite. The work by [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF] provides a flexible model called max-mixture. Let X be a max-stable process, with extremal coefficient Θ(h) and exponent measure function V X h . Let Y be an inverse max-stable process with tail dependence coefficient η(h) and exponent measure function V Y h . Assume that X and Y are independent and each of them has Fréchet margin. Let a ∈ [0, 1] and define

Z(s) = max{aX(s), (1 -a)Y (s)}, s ∈ S.
Z has unit Fréchet marginals. Its bivariate distribution function is given by (2.8)

P Z(s) ≤ z 1 , Z(s+h) ≤ z 2 = e -aV X h (z1,z2) e -(1-a) z 1 +e -(1-a) z 2 -1+e -V Y h (ga(z1),ga(z2)) ,
where g a (z) = g( z 1-a ). Its bivariate survivor function satisfies

P Z(s) > z, Z(t) > z ∼ a{2 -Θ(h)} z + (1 -a) 1/η(h) z 1/η(h) + O(z -2 ), z → ∞.
If h * = inf{h : Θ(h) = 0} < ∞, then Z is asymptotically dependent up to distance h * and asymptotically independent for larger distances. See [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] for more details.

Of course, if a = 0 then Z is indeed an inverted max-stable process. If a = 1 then Z is a max-stable process. Moreover (2.9)

χ(h) = a(2 -Θ(h))
and

(2.10)

χ(h) = 1 [h * <h] (h) + (2η(h) -1)1 [h * ≥h) .

Spatial risk measures.

Consider a spatial process X := {X(s), s ∈ S}, S ⊂ R 2 . We use the definition of risk measures proposed in [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] and [START_REF]Spatial risk measures and applications to max-stable processes[END_REF]. Given a damage function D : R 2 -→ R + , and A ∈ B(R d ), the normalized aggregate loss function on A is

L(A, D) = 1 |A| A D(s)ds,
where |A| stands for the volume of A. The quantity A D(s)ds represents the aggregated loss over the region A. Therefore the function L(A, D) is the proportion of loss on a A.

Definition of spatial risk measures. In this paper, we work with unit

Fréchet margin processes and thus (X -u) + as no finite expectation nor variance. The risk measure considered in [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] is not suitable. In [START_REF]Spatial risk measures and applications to max-stable processes[END_REF], the damage function 1 {X>u} is considered and the subsequent risk measure is computed for Smith, Schlater and the so-called tube processes. This damage function does not take into account the behaviour of the process over the threshold u, this is why, we did not consider it. We shall consider the damage function

D ν X (s) = |X(s)| ν , for 0 < ν < 1
2 . This type of damage function is used e.g. in analyzing the negative effects due to the wind speed (see [START_REF]Natural catastrophes 2012 analyses, assessments, positions 2013 issue[END_REF] for more details). In [START_REF]Spatial risk measures and applications to max-stable processes[END_REF], the risk measure associated to D ν X has been computed for Smith processes. Since we work with stationary processes, the expectation of the normalized loss function do not take into account the dependence structure. As in [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] and [START_REF]Spatial risk measures and applications to max-stable processes[END_REF], we shall focus on its variance.

R 1 (A, D X )} = Var L(A, D X ) . If X is a spatial process with unit Fréchet marginal distributions, R 1 (A, D ν X ) is well defined provided that 0 < ν < 1 2 . Remark that (3.1) R 1 (A, D X )} = 1 |A| 2 A×A Cov D X (s), D X (t) dsdt.
3.2. Axiomatic properties of spatial risk measures. Several authors such as [START_REF] Artzner | Coherent measures of risk[END_REF], [START_REF]Higher moment coherent risk measures[END_REF] and [START_REF] Tsanakas | Risk measures and theories of choice[END_REF] presented an axiomatic setting for univariate risk measures. In [START_REF]Spatial risk measures and applications to max-stable processes[END_REF] a first set of axioms for risk measures in spatial context is considered for the damage functions: D X (s) = 1 {X(s)>u} , D X (s) = X(s) ν where X is a max-stable process. In [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] the damage function D X (s) = (X(s) -u) + for Gaussian processes has been investigated.

Let us recall the mains axioms proposed in [START_REF]Spatial risk measures and applications to max-stable processes[END_REF] and [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] for the real valued spatial risk measure R 1 (A, D). Axioms 1. and 4. below have been introduced in [START_REF]Spatial risk measures and applications to max-stable processes[END_REF], and studied for some max-stable processes. Definition 1. Let A ⊂ R 2 be a region of the space.

(1) Spatial invariance under translation

Let A + v ⊂ R 2 be the region A translated by a vector v ∈ R 2 . Then for v ∈ R 2 , R 1 (A + v, D) = R 1 (A, D). (2) Spatial anti-monotoncity Let A 1 ,A 2 ⊂ R 2 , two regions such that |A 1 | ≤ |A 2 |, then R 1 (A 2 , D) ≤ R 1 (A 1 , D). (3) Spatial sub-additivity Let A 1 ,A 2 ⊂ R 2 be two regions disjointed, then R 1 (A 1 ∪A 2 , D) ≤ R 1 (A 1 , D)+ R 1 (A 2 , D). (4) Spatial super sub-additivity Let A 1 ,A 2 ⊂ R 2 be two regions disjointed, then R 1 (A 1 ∪A 2 , D) ≤ min i=1,2 [R 1 (A i , D)]. (5) Spatial homogeneity Let λ > 0 and A ⊂ R 2 then R 1 (λA, D) = λ k R 1 (A, D), that is R 1 is homogenous of order k, where λA is the set {λx, x ∈ A}.
In [START_REF]Spatial risk measures and applications to max-stable processes[END_REF] the invariance by translation, the monotonicity and super sub-additivity in the case where A 1 , A 2 are either disks or squares is proved for max-stable processes for the damage function 1 {X>u} and for the damage function X ν in the case of the Smith process. While in [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] the invariance by translation and sub-additivity is proved for any processes provided that D X admits an order 2 moment. The anti-monotonicity for disks and squares is proved for the damage function (X -u) + with X a Gaussian process. We shall study further the properties of R 1 (A, D X ) for max-mixture processes.

4. Risk measures for max-mixture processes.

Let X an isotropic and stationary process, with unit Fréchet margin, let 0 < ν < 1/2 be a fixed.

General forms for R

1 (A, D ν X ).
The following result shows that the computation of R 1 (A, D X ) may reduce to smaller dimension integral. It has been proved in [START_REF] Koch | Tools and models for the study of some spatial and network risks:application to climate extremes and contagion in france[END_REF] for Smith models. Following the lines of its proof, it remains valid provided that the damage function X ν has an order 2 moment (see Theorem 3.3 in [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF]).

Let f disk (•, R) and f square (•, R) be the density of the distance between two points randomly chosen in a disk of radius R and a square of side R respectively. We have (see [START_REF] Moltchanov | Distance distributions in random networks[END_REF]):

f disk (h, R) = 2h R 2 2 π arccos h 2R - h πR 1 - h 2 4R 2
, and

f square (h, R) = 2πh R 2 - 8h 2 R 3 + 2h 3 R 4 where b = h 2 R 2 . Lemma 4.1.
Let X := {X(s), s ∈ S} be an isotropic and stationary spatial process such that the damage function D X has finite order 2 moment. Let Q(h) = Cov D X (s), D X (s + h) . Consider A ⊂ R 2 a disk of radius R, we have:

(4.1) R 1 (A, D X ) = Var L(A, D X ) = 2R h=0 Q(h)f disk (h, R)dh,
Consider A ⊂ R 2 a square of side R, we have:

(4.2) R 1 (A, D X ) = Var L(A, D X ) = √ 2R h=0 Q(h)f square (h, R)dh,
In what follows, results are written for square regions A, but the results hold for disks as well.

Remark 1. Properties of moments of Fréchet distributions give that if X has unit Fréchet marginal distributions,

E(L(A, D ν X )) = Γ(1 -ν).
Proposition 4.2. Consider X := {X(s), s ∈ S} an isotropic and stationary spatial process with unit Fréchet margin F and pairwise distribution function G X h = P(X(s) ≤ x 1 , X(s + h) ≤ x 2 ). Let A be a square of side R. We have

(4.3) R 1 (A, D ν X ) = √ 2R h=0 Q(h, ν)f square (h, R)dh, with Q(h, ν) = Cov D ν X (s), D ν X (s + h) ; (4.4) Q(h, ν) = ∞ 0 ∞ 0 G X h (x 1/ν 1 , x 1/ν 2 ) -F (x 1/ν 1 )F (x 1/ν 2 ) dx 1 dx 2
or equivalently

(4.5) Q(h, ν) = ν 2 ∞ 0 ∞ 0 x ν-1 1 x ν-1 2 G X h (x 1 , x 2 ) -F (x 1 )F (x 2 ) dx 1 dx 2 .
Proof. Since X is a non negative process, the result follows directly from Hoeffding's identity ( [START_REF] Hougaard | Analysis of multivariate survival data[END_REF] and [START_REF] Sen | The impact of wassily hoeffding's research on nonparametrics[END_REF]):

Cov D ν X (s), D ν X (s + h) = R 2 + P X(s) ν ≤ x 1 , X(s + h) ν ≤ x 2 -P X(s) ν ≤ x 1 P X(s + h) ν ≤ x 2 dx 1 dx 2 = ν 2 R + x ν-1 1 x ν-1 2 P X(s) ≤ x 1 , X(s + h) ≤ x 2 -P X(s) ≤ x 1 P X(s + h) ≤ x 2 dx 1 dx 2 .
4.2. Explicit form for R 1 (A, D ν X ) for TEG max-stable process X. Equation (4.3) shows that, if A is either a disk or a square, the computation of R 1 (A, D ν X ) reduces to the integration of Q(h, ν)f square (resp. Q(h, ν)f disk ). In [START_REF] Koch | Tools and models for the study of some spatial and network risks:application to climate extremes and contagion in france[END_REF], the computation of Q(h, ν)f square for the Smith model has been done. In that case, the computation of R 1 (A, D ν X ) is reduced to a one dimensional integration. In this section, we do the computation for a TEG model. Corollary 4.3. Let X := {X(s), s ∈ S} be a truncated extremal Gaussian TEG max-stable process with unit Fréchet margin, correlation function ρ and truncated parameter r. For 0 < ν < 1/2, we have

Q(h, ν) = +∞ 0 w ν Γ(2(1 -ν))T 2 (w, h)T 1 (w, h) 2(ν-1) + Γ(1 -2ν)T 3 (w, h)T 1 (w, h) 2ν-1 dw -Γ(1 -ν) 2 where, (4.6) T 1 (w, h) = w + 1 w 1 - α(h) 2 1 -K(w, h) ; T 2 (w, h) = 1 - α(h) 2 1 -K(w, h) - α(h)(ρ(h) + 1)(1 -w) 2K(w, h)(w + 1) 2 × 1 w 2 - α(h) 2w 2 1 -K(w, h) - α(h)(ρ(h) + 1)(w -1) 2wK(w, h)(w + 1) 2 ; (4.7) (4.8) T 3 (w, h) = α(h) (ρ(h) + 1) K(w, h)(w + 1) 3 - (ρ(h) + 1) 2 (w -1) 2 2K(w, h) 3 (w + 1) 5 ; K(w, h) = 1 - 2w(ρ(h) + 1) (w + 1) 2 1/2 and α(h) = {1 -h 2r } + . Proof. We have, Cov D ν X (s), D ν X (s + h) = E D ν X (s)D ν X (s + h) -E[D ν X (s)] 2 . From Remark 1, E[D ν X (s)] = Γ(1 -ν). Moreover, E D ν X (s)D ν X (s + h) = ∞ 0 ∞ 0 x ν 1 x ν 2 f (X(s),X(s+h)) (x 1 , x 2 )dx 1 dx 2 ,
where f (X(s),X(s+h)) (x 1 , x 2 ) is the bivariate density function of the TEG model. It rewrites:

E[D ν (s)D ν (s + h)] = +∞ 0 +∞ 0 u 2ν+1 w ν f (u, uw)dudw.
The bivariate density function of a TEG model is given by

f (X(s),X(s+h)) (u, uw) = 1 u 4 T 2 (w, h) + 1 u 3 T 3 (w, h) e -1 u T1(w,h)
where T 1 (w, h), T 2 (w, h) and T 3 (w, h) are given in (4.6), (4.7) and (4.8). Therefore

E[D ν (s)D ν (s + h)] = +∞ 0 w ν T 2 (w, h) +∞ 0 u 2ν-3 e -1 u T1(w,h) du + T 3 (w, h) +∞ 0 u 2ν-2 e -1
u T1(w,h) du dw.

Moment properties of Fréchet distributions give

+∞ 0 u 2ν-3 e -1 u T1(w,h) du = 1 T 1 (w, h) .µ (2ν-1) ,
with µ (2ν-1) the moment of order k = (2ν -1) of a Fréchet distribution. In the same way, we get

+∞ 0 u 2ν-2 e -1 u T1(w,h) du = T 1 (w, h) (2ν-1) Γ(1 -2ν).
Then,

E D ν X (s)D ν X (s + h) = +∞ 0 w ν T 2 (w, h)T 1 (w, h) 2(ν-1) Γ2(ν -1) + T 3 (w, h)T 1 (w, h) (2ν-1) Γ(1 -2ν) dw,
and the result follows.

Corollary 4.3 shows that the risk measure for a TEG process may be computed efficiently, since it reduces to a one dimensional integration involving a Gamma function.

4.3.

Behavior of R 1 (λA, D ν X ) with respect to λ for max-mixture processes. In what follows, we consider an isotropic and stationary max-mixture spatial process with unit Fréchet margin F . We denote X and V X h the process and the exponent measure function corresponding to the max-stable part and Y and V Y h the process and the exponent measure function corresponding to the inverse max-stable process Y . Let a ∈ [0, 1], Z = max(aX, (1 -a)Y ). We shall study the behavior of R 1 λA, D ν Z with respect to λ. Of course, the case a = 1 gives results for maxstable processes and a = 0 gives results for inverse max-stable processes. Recall that the bivariate distribution function is given by

G Z h (x 1 , x 2 ) = e -aV X h (x1,x2) e -(1-a) x 1 + e -(1-a) x 2 -1 + e -V Y h (ga(x1),ga(x2)) ,
where g(z) = -

1 log(1-e -1 z )
and g a (z) = g( z 1-a ). Lemma 4.1 and Proposition 4.2 are a keystone to describe the behaviour of R 1 λA, D ν Z . As in Lemma 3.4 in [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF], we get for any λ > 0:

(4.9) R 1 (λA, D ν Z ) = √ 2R h=0 f square (h, R)Q(λh, ν) dh.
Corollary 4.4. Let Z be an isotropic and stationary max-mixture spatial process as above. Assume that the mappings h → V X h (x 1 , x 2 ) and h → V Y h (x 1 , x 2 ) are non decreasing for any (x 1 , x 2 ) ∈ R 2 + . Let A ⊂ S be either a disk or a square, then the mapping λ → R 1 (λA, D ν Z ) is non-increasing. Proof. We use (4.9) and from Proposition 4.2,

Q(h, ν) = ν 2 ∞ 0 ∞ 0 x ν-1 1 x ν-1 2 G Z h (x 1 , x 2 ) -F (x 1 )F (x 2 ) dx 1 dx 2 . Since h → V X h (x 1 , x 2 ) and h → V Y h (x 1 , x 2 ) are non decreasing, h → G Z h (x 1 , x 2
) is non increasing and the result follows.

Remark 2. For a spatial max-stable or inverse max-stable process X, the fact that h → V X h (x 1 , x 2 ) is non decreasing implies that the dependence between X(t) and X(t + h) decreases as h increases, which seems reasonable in applications. On another hand, if in addition, V X h (x 1 , x 2 ) goes to 1 x1 + 1 x2 as h goes to infinity, this means that X(t), X(t + h) tend to behave independently as h goes to infinity. Corollary 4.5. Let Z be an isotropic and stationary max-mixture spatial process as above. Assume that the mappings h → V X h (x 1 , x 2 ) and h → V Y h (x 1 , x 2 ) are non decreasing for any (x 1 , x 2 ) ∈ R 2 + . Moreover, we assume that

V X h (x 1 ; x 2 ) -→ 1 x 1 + 1 x 2 as h → ∞ and V Y h (x 1 , x 2 ) -→ 1 x 1 + 1 x 2 as h → ∞ ∀x 1 , x 2 ∈ R + .
Let A ⊂ S be either a disk or a square, we have

lim λ→∞ R 1 (λA, D ν Z ) = 0.
If there exists V 0 (resp. V 1 ) an exponent measure function of a non independent max-stable (resp. inverse max-stable) bivariate random vector, such that

V X h -→ V 0 (resp. V Y h -→ V 1 ) as h → ∞, then lim λ→∞ R 1 (λA, D ν Z ) > 0.
Proof. In the case of A a square of side R, we use

Q(h, ν) = ν 2 ∞ 0 ∞ 0 x ν-1 1 x ν-1 2 G Z h (x 1 , x 2 ) -F (x 1 )F (x 2 ) dx 1 dx 2 . If V W h (x 1 ; x 2 ) is non decreasing to 1 x1 + 1 x2 as h → ∞ for W = X or W = Y , then G Z h (x 1 , x 2
) is non increasing to F (x 1 )F (x 2 ) and we conclude by using the monotone convergence theorem.

Corollary 4.6. Let Z be an isotopic and stationary max-mixture as above. Assume

that h → V W h (x 1 , x 2 ) is non increasing, with W = X or W = Y . Let A 1 and A 2 be either disks or squares such that |A 1 | ≤ |A 2 | then R 1 (A 2 , D ν Z ) ≤ R 1 (A 1 , D ν Z ).
Proof. Since the risk measure R 1 (A, D ν Z ) is invariant by translation, we may assume that A 1 = λA 2 for some λ ≥ 1. Then, Equation (4.9) gives the result.

Numerical study

In this section, we will study the behavior of the spatial covariance damage function and its spatial risk measure corresponding to a stationary and isotropic max-stable, inverse max-stable and max-mixture processes. We shall use the correlation functions introduced in [START_REF] Abrahamsen | A review of gaussian random fields and correlation functions[END_REF].

(1) Spherical correlation function:

(3) Exponential correlation functions:

ρ exp θ (h) = exp - h θ , (4)
Gaussian correlation functions:

ρ gau θ (h) = exp - h θ 2 ;
(5) Matérn correlation function:

ρ mat (h) = 1 Γ(κ)2 κ-1 (h/θ) κ K κ (h/θ),
where Γ is the gamma function, K κ is the modified Bessel function of second kind and order κ > 0, κ is a smoothness parameter and θ is a scaling parameter. 5.1. Analysis of the covariance damage function Q(h, ν). The covariance damage function plays a central role in the study of the risk measure R 1 (A, D ν X ). 5.1.1. Analysis of Q(h, ν) for max-stable processes. We study the behavior of Q(h, ν) and R 1 (λA, D ν X ) for X a TEG spatial max-stable process, with trunacted parameter r, non-negative correlation function ρ and correlation length θ. We shall denote by Q θ,r (h, ν) the covariance damage function in order to emphasize the dependence of the parameters. Five different models with different correlation functions (exponential, Gaussian, spherical, cubic and Matern) introduced above are considered.

The behavior of Q θ,r (h, ν) is shown in Figure 1.(a). We set the power coefficient ν = 0.20, r = 0.25 and θ = 0.20. We have that, Q θ,r (h, ν) = 0 for any h ≥ 2r; the decreasing speed changes according to the different dependence structures.

For the behavior of D ν Y (s), s ∈ S with respect to θ is shown in Figure 1(b). Figure 1(c) shows the behavior of Q θ,r (h, ν) with respect to the truncated parameter r. We set ν = 0.20, h = 0.25 and θ = 0.20. Finally, we study the behavior of the spatial damage covariance function with respect to power coefficient ν. We set h = 0.25, θ = 0.20 and r = 0.25. Figure1. (d) shows that the covariance between the damage functions D ν Y (•) and D ν Y (• + h) increases with ν.

Remark 3. The global behavior of Q θ,r (h, ν) for an inverse TEG is the same as for the TEG with the same parameters. 5.1.2. Analysis of Q(h, ν) for max-mixture processes. Max-mixture models with TEG max-stable part, denoted X and inverse TEG for the inverse max-stable part -denoted by Y -cover all possible dependence structures in one model (asymptotic dependence at short distances, asymptotic independence at intermediate distances and independence at long distances). We have simulated five max-mixture models according to the correlation functions above, X and Y have the same correlation functions with different correlation lengths. r X and r Y denote the respective truncation parameter of X and Y , ρ X and ρ Y denote the respective correlation functions of X and Y , θ X and θ Y denote the respective correlation length. The mixing parameter is denoted by a.

We set the parameters a = 0.5, r X = 0.15, θ X = 0.10, r Y = 0.35, θ Y = 0.3 and finally ν = 0.2. In this model, the damage functions D ν Y (•) and D ν Y (• + h) are asymptotically dependent up to distance h < 2r X . The decreasing speed depends on the correlation function, as shown in Figure 2. Figure 3 shows the behavior of Q(h, ν) with respect to each parameter. When it is not varying, each parameter is fixed to a = 0.5, h = 0.25, ν = 0.2, θ X = 0.1, θ Y = 0.3, r X = 0.15 and r Y = 0.35. Graph (a) shows the behavior of Q with respect to the mixing parameter a. The graphs from (b) to (f) shows the behavior of Q with respect to the other parameters. The behavior is the same as for max-stable processes.

5.2. Numerical computation of R 1 (A, D ν ). In this study, we compute R 1 (A, D ν ) for different max-stable processes X, inverse max-stable processes Y and maxmixture processes Z. We considered X a TEG with parameters r X and θ X , Y a Smith process with parameter σ 2 Y . The process Z is a max-mixture between X and Y . Max-stable and inverse max-stable models are achieved for a = 1 and a = 0, respectively. We compute R 1 (A, D ν ) using (4.3) and (4.5) i.e. a 3 dimensional integration. For these models, the reduction to a one dimensional integration seem not possible. We shall compare this computed value with the Monte Carlo estimation obtained by simulating the process Z. In this simulation study, the TEG has parameters: r X = 0.25, non-negative exponential correlation function with θ X = 0.20. The inverse max-stable Y , is given by a Smith max-stable process Y with σ 2 Y = 1. The process Z is simulated with n = 50 locations on a grid over a square A = [0, 1] 2 . We set the power coefficient ν := {0.05, 0.15, 0.25, 0.35, 0.40} and mixing parameter a := {0, 0.25, 0.5, 0.75, 1}.

L j (A, D + Z,u ) = 1 |A| R n -1 2 n-1 i=1 Z * (s ij ) j = 1, ..., m, where, Z * (s ij ) = |Z(s ij )| ν E M 1 [L(A, D ν Z )] = 1 m m j=1 L j (A, D ν Z ) and (5.1) Var M 1 (L(A, D ν Z )) = 1 m -1 m j=1 (L j (A, D ν Z ) -E M 1 [L(A, D ν Z )]) 2 .
Boxplots in Figure 4. represent the relative errors over 100 (M1) simulations with respect to the 3 dimensional integration. It shows that the considered risk measures are hardly estimated by Monte Carlo for ν greater than 0.30. Let us emphasize that in the 3 dimensional integration, we used (4.5). Using (4.4) creates numerical issues when ν approaches 0.4.

5.3.

Behavior of R 1 (λA, D ν X ). We are going to study the behavior of R 1 (λA, D ν X ) with respect to λ for A = [0, 1] 2 a square and several models. We fixed ν = 0.20, and a = 0.50 for max-mixture models and also we will evaluate R 1 (λA, D ν X ) with respect to the mixing parameter a.We considered two models for X: TEG with r = 0.250 and non-negative exponential correlation function with correlation length θ = 0.20; Smith with σ 2 = 0.6. We considered two inverse max-stable processes Y : Inverse TEG and inverse Smith. The process Z is the max mixture Z = max(aX, (1-a)Y ). The different chosen parameters are listed below.

• MM1: X is TEG with the parameters as for the TEG max-stable process above and Y is inverse TEG with r Y = 0.45 and non-negative exponential correlation function with correlation length θ = 0.40. • MM2: X is TEG max-stable with the same parameters as for MM1 and Y is inverse Smith with σ 2 Y = 0.8. Figure 5 is devoted to max-stable and inverse max-stable processes (no mixture). It shows that R 1 (λA, D ν X ) for max-stable and inverse max-stable processes are very similar. Their behavior mimics also the one of χ(h) in the max-stable case, or χ(h) in the inverse max-stable case. In this picture, we have chosen h = 0.3. Figure 6.(a) shows the behavior of R 1 (λA, D ν X ) for the max-mixture model MM1. It shows a relatively high value for R 1 (λA, D ν X ) up to 0.3λ < 2r X . Figure 6.(b) is devoted to the model MM2. The global behavior is the same for the two models. We remark that the rupture parameter r Y is hardly identified on these graphs. Figure 6.(b) shows the behavior of R 1 (λA, D ν X ) with respect to the max-mixture model MM2. We can see the same behavior of the asymptotic dependence part in MM1 when 0.3λ < 2r X and the decrease to zero from 0.3λ ≥ 2r X . The speed of decrease to zero depends the chosen model.

Conclusion

We have developped the study of the risk measure R(A, D ν ) for spatial processes allowing asymptotic dependence and asymptotic independence. This risk measure takes into account the spatial dependence structure over a region. It satisfies the axioms from [START_REF] Ahmed | Spatial risk measure for gaussian processes[END_REF] and [START_REF]Spatial risk measures and applications to max-stable processes[END_REF] for isotropic and stationary max-mixture processes. A simulation study emphasized the behavior of the risk measure with respect to the various parameters. Finally, the sensitivity of spatial risk measures with different dependence structures is studied for two different models.

Figure 1 .

 1 Figure 1. shows the behavior of Q θ,r (h, ν) with respect to the power coefficient ν, the correlation length θ, the distance h and truncated parameter r. Plain lines correspond to TEG and dashed lines correspond to inverse TEG. Five non-negative correlation functions (exponential, Gaussian, spherical, cubic and Matérn with κ = 1) have been examined. The graphs (a), (b) ,(c) and (d) show the behavior of Q •,• (•, •) with respect to: (a) the distance h, when ν = 0.2 ,θ = 0.2 and r = 0.25; (b) the correlation length θ, when ν = 0.2, r = 0.25 and h = 0.25; (c) the truncated parameter r, when ν = 0.2, θ = 0.20 and h = 0.25; (c) the power coefficient ν, when θ = 0.20, r = 0.25 and h = 0.25.

Figure 2 .

 2 Figure 2. shows the behavior of Q(h, ν) with respect to the distance h. Five non-negative correlation functions (exponential, Gaussian, spherical, cubic and Matérn with κ = 1) have been examined when a = 0.5, ν = 0.2 and X is a TEG max-stable process with θ X = 0.15 and r X = 0.10; Y is an inverted TEG process with θ Y = 0.35 and r Y = 0.30.

Figure 3 .

 3 Figure 3. (a) shows the behavior of Q(h, ν) with respect to the mixing parameter a, the power coefficient ν, the correlation lengths θ X , θ Y , and truncation parameters r X , r Y . Five non-negative correlation functions (exponential, Gaussian, spherical, cubic and Matérn with κ = 1) have been examined. For a = 0.5, h = 0.25, ν = 0.2, θ X = 0.1, θ Y = 0.3, r X = 0.15 and r Y = 0.35, the graphs (a),(b),(c),(d),(e) and (f) show the behavior of Q(•, •) with respect to: (a) the mixing parameter a; (b) the power coefficient ν; (c) the truncation parameter r X ; (d) the truncated parameter r Y ; (f) the correlation length θ X ; (e) the correlation length θ Y .

Figure 4 .

 4 Figure 4. The boxplots represent the relative errors of the Monte Carlo estimation of Var(L(A, D ν Z )) with respect to the 3 dimensional integration for different power coefficient ν := {0.05, 0.15, 0.25, 0.35, 0.40} and mixing parameter a := {0, 0.25, 0.5, 0.75, 1} with parameters r X = 0.25 and θ X = 0.20 corresponding to the max-stable process X and with σ 2 = 1 corresponding to the inversed Smith process Y over a square A = [0, 1] 2 .

Figure 5 .

 5 Figure 5. The graphs represent the behavior of R 1 (λA, D ν X ) with respect to λ for ν = 0.20, a square A = [0, 1] 2 and the corresponding to tail and lower tail dependence coefficients. Four models are considered:(a) TEG model with truncated parameter r X = 0.25 and exponential correlation function with correlation length θ X = 0.20; (b) inverse TEG max-stable with the same parameters as in (a); (c) Smith max-stable process with σ 2 = 0.6; (d) inverse Smith max-stable process with the same parameters as in (c). Finally the graphs (1),(2),(3) and (4) represent the tail and lower tail dependence coefficients corresponding to each model receptively, h = 0.3.

Figure 6 .

 6 Figure 6. shows the behavior of R 1 (λA, D ν X ), χ(h) and χ(h) for two max-mixture models.

Figure 7 .

 7 Figure 7. shows the behavior of R 1 (λA, D ν X ) with respect to mixing parameter a for two max-mixture models : (a) MM1 model; (b) MM2 model.

Acknowledgements: This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


M Ahmed

V Maume-Deschamps

P Ribereau

C Vial

A SEMI-PARAMETRIC ESTIMATION FOR MAX-MIXTURE SPATIAL PROCESSES

Keywords: Spatial dependence measures, asymptotic dependence/independence, max-stable process, max-mixture, madogram

We propose a semi-parametric estimation procedure in order to estimate the parameters of a max-mixture model as an alternative to composite likelihood estimation. This procedure uses the F-madogram. We propose to minimize the square difference between the theoretical F-madogram and an empirical one. We evaluate the performance of this estimator through a simulation study and we compare our method to composite likelihood estimation. We apply our estimation procedure to daily rainfall data from East Australia.

Introduction

One of the main characteristic of environmental or climatic data is their spatial dependence. The dependencies may be strong even for large distances as observed for heat waves or they may be strong at short distances and weak at larger distances, as observed for cevenol rainfall events. Many dependence structures may arise, for example, asymptotic dependence, asymptotic independence, or both. Max-mixture models as defined in [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF] are a mixture between a max-stable process and an Asymptotic Independent (AI) process. These kind of models may be useful to fit e.g. rainfall data (see [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF]).

The estimation of the parameters of these processes remains challenging. The usual way to estimate parameters in spatial contexts is to maximize the composite likelihood. For example, in [START_REF] Padoan | Likelihood-based inference for max-stable processes[END_REF], [START_REF] Davison | Geostatistics of extremes[END_REF] and many others, the composite likelihood maximization is used to estimate the parameters of max-stable processes. In [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] and [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF], it is used to estimate the parameters of max-mixture processes. Nevertheless, the estimation remains unsatisfactory in some cases as it seems to have difficulties estimating the AI part. So that, an alternative should be welcomed. We propose a semi-parametric estimation procedure as an alternative to composite likelihood maximization for max-mixture processes. Our procedure is a least square method on the F -madogram. That is, we minimize the squared difference between the theoretical F -madogram and the empirical one. Some of literature deals with semi-parametric estimation in modeling spatial extremes. For example, [START_REF] Northrop | An efficient semiparametric maxima estimator of the extremal index[END_REF] and [START_REF] Aghakouchak | Semi-parametric and parametric inference of extreme value models for rainfall data[END_REF] provided semi-parametric estimators of extremal indexes. In [START_REF] Buhl | Semiparametric estimation for isotropic max-stable space-time processes[END_REF], another semi-parametric procedure to estimate model parameters is introduced. It is based a non linear least square method based on the extremogram for isotropic space-time Browen-Resnick max-stable processes. The semi-parametric procedure that we propose in this study is close to that article. Section 2 is dedicated to the main tools used in this study; it contains definitions of some spatial dependence measures and of different dependence structures (asymptotic dependence/independence and mixture of them). In Section 3, we calculate an expression for the F -madogram of max-mixture models. Section 4 is devoted to estimation procedures of the parameters of max-mixture processes. We prove that our least square-madogram estimation is consistent, provided that the parameters are identified by the F -madogram. In Section 5, simulation study is conducted, it allows us to evaluatere the performance of the estimation procedure. Section 6 is devoted to an application on rainfall real data from East Australia. Finally, concluding remarks are discussed in Section 7.

Main tools used in the study

Throughout this study, the spatial process X := {X(s), s ∈ S}, S ∈ R d is assumed to be strongly stationary and isotopic. We recall some basic facts and definitions related to spatial extreme theory.

2.1. Dependence measures. The extremal dependence behavior of spatial processes may be described by several coefficient / measures. The upper tail dependence coefficient χ (see [START_REF] Bacro | Measuring and modelling multivariate and spatial dependence of extremes[END_REF][START_REF] Sibuya | Bivariate extreme statistics, i[END_REF]) is defined for a stationary spatial process X on S ⊂ R 2 with margin F by: (2.1)

χ(h) = lim u→1 -P F (X(s + h)) > u|F (X(s)) > u . The process is said AI (resp. AD) if for all h ∈ S χ(h) = 0 (resp. χ(h) = 0).
For any h the coefficient χ(h) can alternatively be expressed as the limit when

u → 1 -of the function defined on S × [0, 1] into [0, 1], by (2.2) χ(h, u) = 2 - log P F (X(s)) < u, F (X(t)) < u log P F (X(s)) < u , for h ∈ S, u ∈ [0, 1[. Such that, χ(h) = lim u→ -1 χ(h, u).
For AI processes, χ cannot reveal the strength of the dependence. This is why the authors in [START_REF] Coles | Dependence measures for extreme value analyses[END_REF], introduced an alternative dependance coefficient called lower tail dependence coefficient χ. Consider for any (s,

s + h) ∈ S 2 (2.3) χ(h, u) = 2 log P F (X(s)) > u log P F (X(s)) > u, F (X(s + h)) > u -1, 0 ≤ u ≤ 1 and χ(h) = lim u→1 χ(h, u).
If χ(h) = 1 for all h, the spatial process is asymptotically dependent. Otherwise, the process is asymptotically independent. Furthermore, if χ ∈]0, 1[ ( resp. ] -1, 0[) the locations s and s + h (for any s) are asymptotically positively associated (resp. asymptotically negatively associated).

Another important measure of dependence is the extremal coefficient which was introduced by [START_REF] Buishand | Bivariate extreme-value data and the station-year method[END_REF][START_REF] Schlather | A dependence measure for multivariate and spatial extreme values: Properties and inference[END_REF].For any s ∈ S and s + h ∈ S and x ∈ R, let

θ F (h, x) = log(P (X(s) < x, X(s + h) < x)) log(P (X(s) < x)) .
θ F is related with the upper tail dependence parameter; indeed if lim x→x F θ F (h, x) = θ(h) exists, we have the following relation (see [START_REF] Bacro | Measuring and modelling multivariate and spatial dependence of extremes[END_REF]):

χ(h) = 2 -θ F (h),
where x F = sup{x|F (x) < 1}. In that case, P (X(s) < x, X(s + h) < x)) may be approximated by F (x) θ F (h) for x large.

This coefficient is particularly useful when dealing with asymptotic dependence, but useless in case of asymptotic independence. To overcome this problem, [START_REF] Ledford | Statistics for near independence in multivariate extreme values[END_REF] proposed a model allowing to gather all the different dependence behaviors. This model has Fréchet marginal laws and for all (s, s + h) ∈ S 2 the pairwise survivor function is given by: (2.4) P X(s) > x, X(s + h) > x = L h (x)x -1/η(h) , as x → ∞, where L h is a slowly varying function and η(h) ∈ (0, 1] is the tail dependence coefficient. This coefficient determines the decay rate of the bivariate tail probability for large x. The interest of this simple modelization, which appears to be quite general, is that the coefficient η(h) provides a measure of the extremal dependence between X(s) and X(x + h). L(x) → 0 (resp. 0 < η(h) < 1 and L(x) → 0), corresponds to asymptotic dependence (resp. asymptotic independence); see [START_REF] Bacro | Measuring and modelling multivariate and spatial dependence of extremes[END_REF][START_REF] Ledford | Statistics for near independence in multivariate extreme values[END_REF]. Finally, it is important to see the relation between η and χ. If equation (2.4) is satisfied, then χ(h) = 2η(h)-1.

Another classical tool often used in geostatistics is the variogram. But for spatial processes with Fréchet marginal laws, the variogram does not exist. We shall use the F -madogram introduced in [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF] which is defined for any spatial process X with univariate margin F and for all (s, t) ∈ S 2

(2.5) ν F (s -t) = 1 2 E|F (X(s)) -F (X(t))|.
2.2. Spatial extreme models. For completeness, we recall definitions on maxstable, inverse max-stable and max-mixture processes.

X i -b n a n d → X, n → ∞,
then X := {X(s), s ∈ S} is a max-stable process [START_REF] De Haan | Spatial extremes: Models for the stationary case[END_REF]. When for all n ∈ N, a n = 1 and b n = 0, the margin distribution of the process X is unit Fréchet, that is for any s ∈ S and x > 0,

F (x) := P (X(s) ≤ x) = exp[-1/x],
in that case, we say that X is a simple max-stable porcess.

In [START_REF] Haan | A spectral representation for max-stable processes[END_REF] it is proved that a max-stable process X can be constructed by using a random process and a Poisson process. This representation is named the spectral representation. Let X be a max-stable process on S. Then there exists {ξ i , i ≥ 1} i.i.d Poisson point process on (0, ∞), with intensity dξ/ξ 2 and a sequence {W i , i ≥ 1} of i.i.d. copies of a positive process W = (W (s), s ∈ S), such that E[W (s)] = 1 for all s ∈ S such that (2.7)

X = d max i≥1 ξ i W i .
The c.d.f. of a simple max-stable process satisfies: (see [START_REF] Beirlant | Statistics of extremes: theory and applications[END_REF], Section 8.2.2.)

(2.8) P X(s 1 ) ≤ x 1 , ..., X(s k ) ≤ x k = exp{-V (x 1 , ..., x k )}, where the function

(2.9) V (x 1 , ..., x k ) = E max =1,...,k W (s ) x .
is homogenous of order -1 and is named the exponent measure. One of the interests of the exponent measure is its interpretation in terms of dependence. In fact, the homogeneity of the exponent measure V implies

(2.10) max{1/x 1 , ..., 1/x k } ≤ V (x 1 , ..., x k ) ≤ {1/x 1 + ... + 1/x k }.
In Inequalities (2.10), the lower (resp. upper) bound corresponds to complete dependence (resp. independence).

For simple max-stable process, the extremal coefficient function Θ, for any pairs of sites (s, s + h) ∈ S 2 is the function Θ defined on S (or in R + in isotropic case) with values in [START_REF] Aghakouchak | Semi-parametric and parametric inference of extreme value models for rainfall data[END_REF][START_REF] Antoniadis | Régression non linéaire et applications[END_REF] by (2.11)

P X(s) ≤ x, X(s + h) ≤ x = exp{-Θ(h)/x}, x > 0 We have (2.12) Θ(h) = E max{W (s), W (s + h)} = V (1, 1) ∈ [1, 2].
If for any h ∈ S, Θ(h) = 1 (resp. Θ(h) = 2), then we have complete dependence (resp. complete independence). The case 1 < Θ(h) < 2, for all h ∈ S corresponds to asymptotic dependence. Remark that, for a simple max-stable process, the coefficients Θ and θ F coincide. Furthermore, it is easy to see the relationship between Θ and χ; see [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF] for any

h ∈ S (2.13) Θ(h) = 2 -χ(h).
In the max-stable case, [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF] gives the relation for all h ∈ S,

(2.14) Θ(h) = 1 + 2ν F (h) 1 -2ν F (h) ,
which appears to be helpful to estimate the extremal coefficient Θ from the empirical madogram. The max-stable process X with pairwise distribution function is given by the following equation, for all (s, s + h) ∈ S 2 , (2.15)

P X(s) ≤ x 1 , X(s + h) ≤ x 2 = exp{-V h (x 1 , x 2 )},
We provide below three examples of well-known max-stable models represented by different exponent measures V .

Smith Model (Gaussian extreme value model) [START_REF] Smith | Max-stable processes and spatial extremes[END_REF] with unit Fréchet margin and exponent measure

(2.16) V h (x 1 , x 2 ) = 1 x 1 Φ τ (h) 2 + 1 τ (h) log x 2 x 1 + 1 x 2 Φ τ (h) 2 + 1 τ (h) log x 1 x 2 ; τ (h) = √ h T Σ -1 h and Φ(•) the standard normal cumulative distribution function. For isotropic case τ (h) = 1 σ h 2 . The pairwise extremal coefficient equals Θ(h) = 2Φ τ (h 2 .
Brown-Resnik Model [START_REF] Kabluchko | Stationary max-stable fields associated to negative definite functions[END_REF] with unit Fréchet margin and exponent measure

V h (x 1 , x 2 ) = 1 x 1 Φ 2γ(h) 2 + 1 2γ(h) log x 2 x 1 + 1 x 2 Φ 2γ(h) 2 + 1 2γ(h) log x 1 x 2 ;
2γ(h) is a variogram and Φ(•) the standard normal cumulative distribution function.

The pairwise extremal coefficient given by

Θ(h) = 2Φ 2γ(h) 2 .
Truncated extremal Gaussian Model (TEG) [START_REF] Schlather | Models for stationary max-stable random fields[END_REF] with unit Fréchet margin and exponent measure (2.17)

V h (x 1 , x 2 ) = 1 x 1 + 1 x 2 1 - α(h) 2 1 -1 -2(ρ(h) + 1) x 1 x 2 (x 1 + x 2 ) 2 .
The extremal coefficient is given by

(2.18) Θ(h) = 2 -α(h) 1 - 1 -ρ(h) 2 1/2 where α(h) = E{|B ∩ (h + B)|}/E[|B|]
and B is a random set which can consider it a disk with fixed radius r. In such a case, α(h) = {1 -h/2r} + (see [START_REF] Davison | Geostatistics of dependent and asymptotically independent extremes[END_REF]) and χ(h) = 0, ∀h ≥ 2r.

2.4. Inverse Max-stable processes. Max-stable processes are either asymptotically Dependent (AS) or independent. This means that they are not useful to model non trivial AI processes.

In, [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF] a class of asymptotically independent processes is obtained by inverting max-stable processes. These processes are called inverse max-stable processes; their survivor function satisfy Equation (2.4). Let X := {X (s), s ∈ S} be a simple max-stable process with exponent measure V . Let g : (0, ∞) → (0, ∞) be defined by g(x) = -1/ log{1 -e -1/x } and X(s) = g(X (s)). Then, X := {X(s), s ∈ S} is an asymptotic independent spatial process with unit Fréchet margin. The d-dimensional joint survivor function satisfies

P X(s 1 ) > x 1 , ..., X(s d ) > x d = exp -V g(x 1 ), ..., g(x d ) . (2.19)
The tail dependent coefficient is given by η(h) = 1/Θ(h), where Θ(h) is the extremal coefficient of the max-stable process X . Moreover, we have

χ(h) = 2/Θ(h) -1.
With a slight abuse of notations, we shall say that V is the exponent measure of X.

2.5. Max-mixture model. In spatial contexts, specifically in an environmental domain, many scenarios of dependence could arise and AD and AI might cohabite. The work by [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF] provides a flexible model called max-mixture. Let X := {X(s), s ∈ S} be a simple max-stable process with extremal coefficient Θ(h) and bivariate distribution function F X , and let Y := {Y (s), s ∈ S} be an inverse max-stable process whose tail dependence coefficient is η(h). Its bivariate distribution function is denoted F Y . Assume that X and Y are independent. Let a ∈ [0, 1] and define (2.20)

Z(s) = max{aX(s), (1 -a)Y (s)}, s ∈ S,
then Z unit Fréchet marginals and its pairwise survivor function satisfies

(2.21) P Z(s) > z, Z(t) > z ∼ a{2 -Θ(h)} z + (1 -a) 1/η(h) z 1/η(h) + O(z -2 ), z → ∞.
The process (Z(s)) s∈S is called a max-mixture process. Assume there exists finite

h * = inf {h : Θ(h) = 0}; then, (2.22) χ(h) = a(2 -Θ(h)) and (2.23) χ(h) = 1 [h * <h] (h) + (2η(h) -1)1 [h * ≥h) .
Remark 2.1. If there exists finite h * = inf {h : Θ(h) = 0}, then Z is asymptotically dependent up to distance h * and asymptotically independent for larger distances.

Of course, if a = 0 then Z is AI and if a = 1 then Z is a simple max-stable process.

In [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] max-mixture processes are studied. The authors emphasize the fact that these models allow asymptotic dependence and independence to be present at a short and intermediate distances. Furthermore, the process may be independent at long distances (using e.g. TEG processes).

F -madogram for max-mixture spatial process

In extreme value theory and therefore for spatial extremes, one of the main concerns is to find a dependence measure that can quantify the dependences between locations. The χ and χ dependence measures are designed to quantify asymptotic dependence and asymptotic independence respectively (see equations (2.22) and (2.23)). Max-mixture processes have been introduced in order to provide both behaviors. We are then faced with the question of finding an adapted tool which would give information on more than one dependence structure.

In [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF], the F -madogram has been introduced for max-stable processes. There exists several definitions of madograms. For example, in [START_REF] Naveau | Modelling pairwise dependence of maxima in space[END_REF], the λ-madogram is considered in order to take into account the dependence information from the exponent measure V h (u, v) when u = v. This λ-madogram has been extended in [START_REF] Fonseca | Generalized madogram and pairwise dependence of maxima over two regions of a random field[END_REF] to evaluate the dependence between two observations located in two disjoint regions in R 2 . [20] adopted a F -madogram suitable for asymptotic independence instead of asymptotic dependence only. Finally, [START_REF] Bacro | Testing the independence of maxima: from bivariate vectors to spatial extreme fields[END_REF] used F-madogram as a test statistic for asymptotic independence bivariate maxima.

Below, we calculate ν F (h) for a max-mixture process. It appears that contrary to χ and χ, it combines the parameters coming from the AD and the AI parts. Proposition 3.1. Let Z be a max-mixture process, with mixing coefficient a ∈ [0, 1]. Let X be its max-stable part with extremal coefficient Θ(h). Let Y be its inverse max-stable part with tail dependence coefficient η(h). Then, the F -madogram of Z is given by

(3.1) ν F (h) = a(Θ(h) -1) a(Θ(h) -1) + 2 - aΘ(h) -1 2aΘ(h) + 2 - 1/η(h) aΘ(h) + (1 -a)/η(h) + 1 β aΘ(h) + 1 (1 -a) , 1/η(h) ,
where β is beta function.

Proof. We have

(3.2) ν F (h) = 1 2 E|F (Z(s)) -F (Z(s + h))|. The equality |a -b|/2 = max(a, b) -(a + b)/2 leads to (3.3) ν F (h) = E max F (Z(s)), F (Z(s + h)) -E F (Z(s)) .
Let M (h) = max F (Z(s)), F (Z(s + h)) , we have:

(3.4) P M (h) ≤ u = P Z(s) ≤ F -1 (u), Z(s + h) ≤ F -1 (u) .
From definition of the max-mixture spatial process Z, we have (3.5)

P Z(s) ≤ z 1 , Z(s+h) ≤ z 2 = e -aV h X (z1,z2) e -(1-a) z 1 +e -(1-a) z 2 -1+e -V h Y (g(z1),g(z2)) ,
where V X (resp.V Y ) corresponding to the exponent measures of X (resp. Y ) and

g(z) = -1/ log(1 -e -(1-a) z
). That leads to

P M (h) ≤ u = u aΘ(h) 2u (1-a) -1 + 1 -u (1-a) 1/η(h) , u ∈ [0, 1]. We deduce that E[M (h)] = 1 0 uf M (h)(u)du = 2a(Θ(h) -1) + 2 a(Θ(h) -1) + 2 - aΘ(h) aΘ(h) + 1 - β aΘ(h)+1 (1-a) , 1/η(h) η(h)(1 -a) aΘ(h)+1 (1-a) + (1/η(h))
.

(3.6) where f M (h) is the density of M (h). Recall that E(F (Z(s))) = 1 2 because F (Z(s)) ∼ U([0, 1]
) and return to equation (3.3) to get equation (3.1).

In the particular cases where a = 1 or a = 0, Proposition 3.1 reduces to known results for max-stable processes (see [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF]) and inverse max-stable processes (see [START_REF] Guillou | Madogram and asymptotic independence among maxima[END_REF]). That is, the F -madogram for a max-stable spatial process is given by

(3.7) 2ν F (h) = Θ(h) -1 Θ(h) + 1 .
and the F -madogram of an asymptotically independent spatial process is given by

(3.8) 2ν F (h) = 1 -η(h) 1 + η(h)
In order to have a comprehensive view of the behavior of ν F (h), we have plotted in Figure 1 below h ν F (h). We have considered two max-mixture models MM1 and MM2 described as follows:

• MM1 max-mixture between a TEG max-stable process X and an inverse Smith max-stable process Y ; • MM2 max-mixture between X as in MM1 and an inverse TEG max-stable process Y . In this Figure and for the two models MM1 and MM2, ν F (h) has two sill one corresponding to X and the second corresponding to Y . This is completely in accordance with the nested variogram concept as presented in [START_REF] Wackernagel | Multivariate nested variogram[END_REF]. In data analysis, these two levels of the sill gives the researcher a hint about whether there is more than one spatial dependence structure in the data. Since the F -madogram expresses with all the model parameters it should be useful for the parameter estimation.

Model inference

This section is devoted to the parametric inference for max-mixture processes. We begin with the presentation of the maximum composite likelihood estimation, then we present the least squares madogram estimation.

Parametric Estimation using Composite Likelihood.

Consider (Z k (s 1 ), . . . , Z k (s D )), k = 1, . . . , N , be N independent copies of a spatial process (Z(s)) s∈S , observed at D locations s 1 , . . . , s D . Composite likelihood inference is an appropriate approach in estimating the parameter models of a spatial process X ( [START_REF] Lindsay | Composite likelihood methods[END_REF][START_REF] Varin | An overview of composite likelihood methods[END_REF]). Asymptotic properties of this estimator has been proved in [START_REF] Davis | Comments on pairwise likelihood in time series models[END_REF]. This approach has been applied successfully to spatial max-stable processes by [START_REF] Davison | Geostatistics of extremes[END_REF] and [START_REF] Padoan | Likelihood-based inference for max-stable processes[END_REF] and is also used to identify the parameters of data exceedances over a large threshold, for example, [START_REF] Bacro | Estimation of spatial max-stable models using threshold exceedances[END_REF] and [START_REF] Thibaud | Threshold modeling of extreme spatial rainfall[END_REF]. In this article, we focus on max-mixture models. Composite likelihood inference for max-mixture processes has been studied in [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] and [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF]. We will compare our madogram based estimation to the composite likelihood estimation described in [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF].

If the pairwise density of Z can be computed and its parameter ψ is identifiable, then it is possible to estimate ψ by maximizing the pairwise weighted log likelihood. For simplicity, we denote

Z k i for Z k (s i ). Let ψ L = max ψ P(ψ),
where (4.1)

P(ψ) = N k=1 D-1 i=1 D j>i w ij log L (Z k i , Z k j ; ψ) =: N k=1 P k (ψ).
where L is the likelihood of the pair (Z k i , Z k j ) and w i,j ≥ 0 is the weight that specifies the contribution for each pair. In [START_REF] Padoan | Likelihood-based inference for max-stable processes[END_REF], it is suggested to In [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF], it is suggested to consider a censor approach of the likelihood, taking into account a threshold. This approach has been adopted in this study. Let G(•, •) be a pairwise distribution function and consider the thresholds u 1 and u 2 ; the likelihood contribution is

L (z 1 , z 2 ; ψ) = ∂ 2 12 G(z 1 , z 2 ; ψ) if z 1 > u 1 , z 2 > u 2 , G(z 1 , z 2 ; ψ) if z 1 ≤ u 1 , z 2 ≤ u 2 ,
In [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF], the censored likelihood is used in order to improve the estimation of the parameters related to asymptotic independence. This censored approach was also applied by [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] for the estimation of parameters of max-mixture processes. In that paper, the replications Z 1 , . . . , Z N of Z are assumed to be α-mixing rather than independent. We denote generically by ψ the parameters of the model. In [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF], it is proved, under some smoothness assumptions on the composite likelihood, that the composite maximum likelihood estimator ψ L for max-mixture processes is asymptotically normal as N goes to infinity with asymptotic variance

G (ψ) = J (ψ)[K(ψ)] -1 J (ψ),
where J (ψ) = E[-∇ 2 P(ψ)], K(ψ) = var(∇P(ψ)). The matrix G (ψ) is called the Godambe information matrix (see [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] and Theorem 3.4.7 in [START_REF] Guyon | Random fields on a network: modeling, statistics, and applications[END_REF]). An estimator J of J (ψ) is obtained from the Hessian matrix computed in the optimization algorithm. The variability matrix K(ψ) has to be estimated too. In our context, we have independent replications of Z and N is large compared with respect to the dimension of ψ. Then, we can use the outer product of the estimation of ψ. Let

K(ψ) = N -1 N k=1 ∇P k ( ψ)∇P k ( ψ)
or by Monte Carlo simulation with explicit formula of P k (ψ) (see section 5. in [START_REF] Varin | An overview of composite likelihood methods[END_REF]). In the case of samples of Z satisfying the α-mixing property, the estimation of K(ψ) can be done using a subsampling technique introduced by [START_REF] Edward | The use of subseries values for estimating the variance of a general statistic from a stationary sequence[END_REF]; this was used in [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF]. Finally, model selection can be done by using the composite likelihood information criterion [START_REF] Varin | A note on composite likelihood inference and model selection[END_REF]:

CLIC = -2 P( ψ) -tr( J -1 K) .
Considering several max-stable models, the one that has the smallest CLIC will be chosen. In [START_REF] Thibaud | Threshold modeling of extreme spatial rainfall[END_REF], the criterion CLIC * = (D -1) -1 CLIC is proposed. It is close to Akaike information criterion (AIC).

4.2.

Semi-parametric estimation using NLS of F-madogram. In this section, we shall define the non-linear least square estimation procedure of the parameters set ψ corresponding to the max-mixture model Z using the F -madogram. This procedure can be considered as an alternative method to the composite likelihood method. Consider Z k , k = 1, . . . , N copies of an isotropic max-mixture process Z with unit Fréchet marginal laws (F denotes the distribution function of a unit Fréchet law). It may be independent copies for example, if the data is recorded yearly (see [START_REF] Naveau | Modelling pairwise dependence of maxima in space[END_REF]) or we shall consider that (Z k ) k=1,...,N satisfies a α-mixing property ( [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF]). Let H be a finite subset of S, J(x, y) = 1 2 |x -y| and Y h,k = J F (Z k (s)), F (Z k (s + h)) , k = 1, .., N and h ∈ H. Therefore, for k = 1, . . . , N , the vectors (Y h,k ) h∈H have the same law and are considered either independent or α-mixing (in k). The main motivation for using the F-madogram in estimation is that it contains the dependence structure information for a fixed h of Y h,k (see Section 3.2 in [START_REF] Bacro | Testing the independence of maxima: from bivariate vectors to spatial extreme fields[END_REF]). In what follows, we make the assumption that the vectors (Y h,k ) h∈H are i.i.d. Note that from the definition of the F -madogram, we have E[Y h,k ] = ν F (h, ψ) where ν F (h, ψ) is the F -madogram of Z with parameters ψ defined in (3.1). If Z has an unknown true parameter ψ 0 on a compact set Ψ ⊂ R d , we rewrite

(4.2) Y h,k = ν F (h, ψ 0 ) + ε h,k .
The vectors (ε h,k ) h∈H are i.i.d errors with E[ε h,k ] = 0 and Var(ε h,k ) = σ 2 h > 0 is finite and unknown. Let

(4.3) L(ψ) = h∈H 1 N k=1,...,N Y h,k -ν F (h, ψ) 2
Any vector ψ M in Ψ which minimizes L(ψ) will be called a least square estimate of ψ 0 :

(4.4) ψ M ∈ argmin ψ∈Ψ L(ψ).
Theorem 4.1. Assume that Ψ ⊂ R d is compact and that ψ → ν F (h, ψ) is continuous for all h ∈ H. We assume that the vectors (Y h,k ) h∈H are i.i.d. Let ( ψ N M ) N ∈N be least square estimators of ψ 0 . Then, any limit point (as N goes to infinity) ψ of (

ψ N M ) N ∈N satisfies ν(h, ψ) = ν(h, ψ 0 ) for all h ∈ H. Remark 4.1. Of course, if ψ (ν(h, ψ))
h∈H is injective, then Theorem 4.1 implies that the least square estimation is consistent, i.e. ψ N M → ψ 0 a.s. as T goes to infinity. In the examples considered below, it seems that the injectivity is satisfied provided |H| ≥ d, but we were unable to prove it.

Proof. We follow the proof of Theorem II.5.1 in [START_REF] Antoniadis | Régression non linéaire et applications[END_REF]. From (4.2), we have, for all

ψ ∈ Ψ L(ψ) = h∈H 1 N k=1,...,N ν F (h, ψ 0 ) + ε h,k -ν F (h, ψ) 2 = h∈H ν F (h, ψ 0 ) -ν F (h, ψ) 2 + 2 N h∈H ν F (h, ψ 0 ) -ν F (h, ψ)   k=1,...,N ε h,k   + h∈H 1 N k=1,...,N ε 2 h,k .
From the law of large numbers, we have 1

N h∈H k=1,...,N ε 2 h,k → h∈H σ 2 h a.s. as N → ∞
and for any h ∈ H, 1 N k=1,...,N ε h,k → 0 a.s.

Therefore,

L(ψ) → h∈H σ 2 h + h∈H ν F (h, ψ 0 ) -ν F (h, ψ) 2 a.s. as N → ∞.
Take a sequence ( ψ N M ) N ∈N of least square estimators, taking if necessary a subsequence, we may assume that it converges to some ψ * ∈ Ψ. Using the continuity of ψ ν F (h, ψ), we have

L( ψ N M ) → h∈H σ 2 h + h∈H ν F (h, ψ 0 ) -ν F (h, ψ * ) 2 a.s. as N → ∞. Since ψ N M is a least square estimator, L( ψ N M ) ≤ L(ψ 0 ) → h∈H σ 2 h . It follows that h∈H ν F (h, ψ 0 ) -ν F (h, ψ * ) 2 = 0
and thus ν(h, ψ * ) = ν(h, ψ 0 ) for all h ∈ H.

The asymptotic normality of the least square estimators should also be obtained by following, e.g., [START_REF] Buhl | Semiparametric estimation for isotropic max-stable space-time processes[END_REF] and using the asymptotic normality of the F -madogram obtained in [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF]. Nevertheless, the calculation of the asymptotic variance will require to calculate the covariances between ν F (h 1 , ψ) and ν F (h 2 , ψ), which is not straightforward.

Simulation study

This section is devoted to some simulations in order to evaluate the performance of the least square estimator and to compare it with the maximum composite likelihood estimator. Recall that ψ M denotes the least square estimator of the parameter vector ψ and ψ L denotes the composite likelihood estimator. 5.1. Outline the simulation experiment. In order to evaluate the performance of the non-linear least square estimator ψ M as defined in (3.1), we have generated data from the model MM1 above. The least square madogram estimator ψ M has been compared with true one ψ 0 and also with parameters estimated by maximum composite likelihood ψ L proposed in [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF][START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF], on the same simulated data. We considered 50 sites randomly and uniformly distributed in the square A = [0, 1] 2 . We have generated N = 1.000 i.i.d observations for each site and replicated this experiment J = 100 times. We have considered several mixing parameters: a := {0, 0.25, 0.5, 0.75, 1}. For the composite likelihood estimator ψ L , we used the censored procedure with the 0.9 empirical quantile of data at each site as threshold u. The fitting of ψ L was done using the code which was used in [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] with some appropriated modifications.

5.2.

Results on the parameters estimate. In Figure 2, we represented the boxplots of the errors, that is ( ψ M -ψ 0 ) and ( ψ L -ψ 0 ) for model MM1. Generally, the estimators above worked well, although the variability in some estimates were relatively large, especially for the asymptotic independence parameters. It also shows some bias in the estimation of the asymptotic independence model parameters.

It is well known that asymptotic independence is difficult to estimate (see [START_REF] Davison | Geostatistics of dependent and asymptotically independent extremes[END_REF]). Therefore, the estimation accuracy of the parameters is very sensitive. On one other hand, the fitting of α(h) which appears in TEG models in (2.17), is delicate and might lead to quite different estimates with different data [START_REF] Davison | Geostatistics of extremes[END_REF]. Another comparison indicator is the root mean square error (RMSE) ( [START_REF] Zheng | Assessing the performance of the independence method in modeling spatial extreme rainfall[END_REF][START_REF] Zheng | Modeling dependence between extreme rainfall and storm surge to estimate coastal flooding risk[END_REF]): let ψ j denote the jth estimation (either least square or composite likelihood estimation), (5.1)

RMSE = J -1 J j=1 ( ψ j -ψ) 2 1/2 ,
The barplot in Figure 3 displays the RMSE for each parameter of MM1 model. We see on these barplots that when a is close to 0 (a = 0; 0.25), the estimator ψ M over-performs the estimator ψ L and vice versa when a ∈ {0.75, 1}. For a = 0.5 the performance of the two estimators seem equivalent.

5.3. Asymptotic normality. We may figure out whether the least square estimator ψ M is asymptotically normal through the graphe of the errors ψ L -ψ 0 . In Figure 4, the graphs represent the distributions of the errors of each estimated parameters of MM1 model for a := {0, 0.5, 1}. We did J = 500 simulation experiments. We can see on Figure 4 that the densities of the errors of the parameters seem close to the shape of centered normal distribution.

Real data example

In this section, we analyze a real data and fit it to some models considered in this study by composite likelihood and LS-madogram procedures. 6.1. Data analysis. We analyzed daily rainfall along the east coast of Australia. We selected 39 locations randomly from this region. The data is daily measured in the period from April to September for 35 years from 1982-2016. The data is available from the Australian Bureau of Meteorology (http://www.bom.gov.au/climate/data/). In order to explore the possibility of anisotropy of the spatial dependence, we used the same test as in [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF]. We divided all data set according to directional sectors (-π/8, π/8], (π/8, 3π/8], (3π/8, 5π/8] and (5π/8, 7π/8], where 0 indicate to north direction. We use the empirical F-madogram ν F (h). The directional loss smoothing of such empirical measure in the Figure 5. (A), shows no evidence of anisotropy.

In Figure 5. (B), the empirical F-madogram is plotted for the whose data set. It seems that asymptotic dependence between the locations is present up to distance 500 km and asymptotic independence could be present at the remaining distance. 6.2. Data fitting. Our interest in this section is to chose a reasonable model for the data. We considered 7 models described below, for each model, the parameters are estimated by LS-madogram and maximum composite likelihood. The selection criteria for LS-madogram estimator ψ M is computed as the following:

M IC := log L( ψ) + (2k(k + 1)/(T -k -1)),
where k is the number of parameters in the model and T is the number of the observations, that is: T = N × |H|, where |H| is the number of observed pairs. With respect to censored composite likelihood estimators we adopted the CLIC selection criteria. The two criteria selected the model MM1 as the best model. We consider the following models:

MM1: max-mixture between asymptotic dependence process represented by TEG max-stable process X with exponential correlation function ρ(h) = exp{-(h/θ X )}, θ x > 0 and B X is a disk of fixed and unknown radius r X . The asymptotic independence is represented by an inverse Brown-Resnik max-stable process Y with variogram 2γ(h) = 2σ MM2: max-mixture between X as in MM1 and an inverse inverse Smith max-stable process Y with τ (h) = h/ √ σ Y .

M1: A TEG max-stable process X specified in MM1. M2: A Brown-Resnik max-stable process X as specified in MM1.

M3: An inverse Brown-Resnik max-stable process Y as specified in MM1. M4: A Smith max-stable process X as specified in M2. M5: A inverse Smith max-stable process Y as specified in M2. For all the considered models, the margin distribution are assumed to be unit Fréchet. Therefore it requires to transform the data set to Fréchet. Most of papers (see for example [START_REF] Bacro | A flexible dependence model for spatial extremes[END_REF] and [START_REF] Wadsworth | Dependence modelling for spatial extremes[END_REF]) use parametric transformations: they fit GEV parameters for each location separately and then transform the data to Fréchet. In this study, we adopted a non-parametric transformation by using the empirical c.d.f.. For censored composite likelihood procedures, we set u = 0.9 and δ = ∞.

We remark that the two Decision Criteria (CLIC and MIC) choose the same model. We would like to emphasize that CLIC and MIC are not comparable, one is related to the composite likelihood while the other one is related to the least squared madogram difference. For the tested models, we would keep the model with the smallest CLIC or the smallest MIC, depending on the used estimation method. Also, the least squared madogram estimation is involves less computations, indeed, the maximum composite likelihood estimation requires to estimate the Godambe matrix in order to compute the CLIC.

Conclusions

The calculation of the F-madogram for max-mixture processes show that it writes with both the AD and the AI parameters. This leads us to propose a semi-parametric estimation procedure using F-madogram ν F (h) as an alternative to composite likelihood. The simulation study showed that the estimation procedure based on ν F (h) performs better than the composite likelihood procedure when the model is near to asymptotic independence. We applied these estimator procedures to real data example. On the considered example, the results obtained by composite likelihood maximization and least squared madogram difference are similar.

Figure 1 .

 1 Figure 1. h ν F (h) for the max-mixture processes models MM1 and MM2. In MM1, X has correlation function ρ(h) = exp(-h/θ X ), r X = 0.25 and fixed radius θ X = 0.2 and Y has Σ = σ Y I d , σ Y = 0.6. In MM2, X has the same setting as in MM1 and Y has ρ Y (h) = exp(-h/θ Y ), θ Y = 0.8 and fixed radius r Y = 1.35. For the two models, we set a = 0.5.

Figure 2 .

 2 Figure 2. Boxplots display ( ψ-ψ) of estimated parameters vector ψ = ( a, r X , θ X , σ Y ) T for the MM1 model by the two estimators ψ M and ψ L . The figures in the first row and from left to right concern the estimator ψ M for a ∈ {0, 0.25, 0.75, 1}, the second row concerns ψ L . We have set, r X = 0.25, θ X = 0.20 and σ Y = 0.6 over a square A = [0, 1] 2 .

Figure 3 .

 3 Figure 3. Barplots display the RMSE of ψ for each estimated parameters ψ = ( a, r X , θ X , σ Y ) T for MM1 and the corresponding two estimators ψ M and ψ L . The bars in the first row and from left to right represent the RMSE of the estimator ψ M when a := {0, 0.25, 0.75, 1}, respectively and the same for the second row for ψ L . We set r X = 0.25, θ X = 0.20 and σ Y = 0.6 over a square A = [0, 1] 2 .

Figure 4 .

 4 Figure 4. Graphs display the densities of the errors between ψ M and ψ 0 for each estimated parameters in the set ψ = (a, r X , θ X , σ Y ) T of MM1 model. The graphs from first row and last one represent the densities of the errors ψ M -ψ 0 when a := {0, 0.5, 1}. We set r X = 0.25, θ X = 0.20 and σ Y = 0.6 over a square A = [0, 1] 2 .

Figure 5 .

 5 Figure 5. Empirical evaluation of ν F (h). The Grey circles represented the empirical value between all pairs. The lines in (A) represent the smoothed value of the empirical of ν F (h) according to directional sectors (-π/8, π/8], (π/8, 3π/8], (3π/8, 5π/8] and (5π/8, 7π/8]. The line in (B) represent the smoothed value of the empirical of ν F (h) for all directions.

Table 1 .

 1 Summary of the fitted models. the distance scale is kilometres. Composite likelihood procedure indicated by CL and LS-madogram procedure indicated by LS with selection criteria SC are CLIC and MIC, respectively.

	Model		a	θ X	r X	θ Y	σ Y	SC
	MM1 CL 0.262	1217.3	1364.5	3102.4	3.457	6807406
		LS 0.259	1285.7	1390.0	5794.8	2.013 1.917034
	MM2 CL 0.248	31.16	70.15	998.84		7924609
		LS 0.185	35.51	48.14	871.19		1.917234
			θ X	r X
	M1	CL	931	307.86				7926261
		LS 1270	255.64				1.945177
			θ X	σ X	θ Y	σ Y
	M2	CL 931.02 3.078663				7926261
		LS 361.36 1.90816				1.96165
	M3	CL			1644.76 2.702282		7918643
		LS			1383.08 1.394928		1.924574
	M4	CL		85.34				8016633
		LS		193.43				1.988753
	M5	CL					256.39	7988838
		LS					334.60	1.929235

Acknowledgements: This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). We also acknowledge the projet LEFE CERISE.