EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 1035-1049 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Asymptotic Dependence Modeling for Spatio-temporal Max-stable Processes D. Barro1,∗, S. P. Nitiéma1, M. Diallo2 1 UFR-ST, Université Ouaga II, 12 BP: 417 Ouagadougou, Burkina Faso 2 FSEG, Université des SSG. BP: 2575 Bamako, République du Mali Abstract. Max-stability is the foundation of multivariate extreme values analysis. This paper investigates the asymptotic dependence modeling of max-stable processes both with spatial and temporal variables. Specifically the paper provides new characterizations of extremal distributions via a dependence measure of the stochastic joint behavior at given locality s and date t. The analytical forms of spatio-temporal asymptotic dependence structures are provided for the main bivariate and trivariate models of max-stable processes. 2010 Mathematics Subject Classifications: 60G70, 62M30, 62H05, 62H11 Key Words and Phrases: Max-stable process, Spatio-temporal process, extreme values , Copulas, Generalized Pareto distributions 1. Introduction In a spatial framework, modeling the extremes of multivariate phenomenas is an im- portant adequate risk management in environment sciences. Indeed, many environmental extremal problems such as hurricanes, floods, droughts, heat waves, sea height, annual maxima and daily rainfall have an inherent spatial character or are time varying events. Likewise a lot of climate change’s problematics and high impact events climatic phenom- enas include a spatial component and can be modeled by extreme values approach. This kind of prospect, such as climate change, have provided modeling technics and spatial tools of extreme event statistics and their characterization are often of fundamental interest. Multivariate extreme values (MEV) theory is often presented in the framework of co- ordinatewise maxima, so the importance of distinction diminishes. Towards a multivariate analogue of Fisher-Tippett we are looking for some sort of multivariate limit distribution for conveniently normalized vectors of multivariate maxima. For an arbitrary index of set T denoting generally a space of time, a random vector Yt = {Yj (t) ; 1 ≤ j ≤ m, t ∈ T} ∗Corresponding author. Email addresses: dbarro2@gmail.com (D. Barro), pnitiema@gmail.com (P. Nitiéma) moudiallo1@gmail.com (M. Diallo) http://www.ejpam.com 1035 c© 2017 EJPAM All rights reserved. D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1036 in Rm is said to be max-stable if, for all n ∈ N, every Yj (t) = (Y (1) j (t) ; . . . ;Y (n) j (t)) is a n-dimensionnal max-stable vector, that is, there exists suitable and time-varying non- random sequences {an (t) > 0} and { bn (t) ∈ Rd } such as 1 an (t) [Mn (t)− bn (t)] f.d.d−→ X (t) ; t ∈ T, (1) where f.d.d−→ denotes the convergence for the finite-dimensional distributions whileMn (t) = max (Xi (t)) 1 ≤ i ≤n ; t ∈ T being the component-wise maxima of the vector X (t). The theory of extreme values is well elaborated in a statistic and mono-site scheme (see Lo et al [10]). Let suppose now that our random variables both depend on the time indexed by {t ∈ T, T 6= 0} ; and are studied on multiple sites and then is also in- dexed by an area index s ∈ S. This leads to study the collections of random vec- tors ( Y s j (t) ) ; j ≥ 0; t ∈ T ; s ∈ S such that for each fixed couple (t, s), the sequence is independent and identically distributed according to a joint cumulative function Gst . Under the assumption that this function is max-stable, every univariate margins Gst,i lies its own domain of attraction and is expressed by on the space of interest S+ ξi,t,s ={ z ∈ R;σi,t,s + ξi,t,s ( yst,i − µi,t,s ) > 0; 1 ≤ i ≤ n } by Gi (yi (s)) =  exp { − [ 1 + ξi (s) ( yi(s)−µi(s) σi(s) )] −1 ξi(si) } if ξi (s) 6= 0 exp { − exp { − ( yi(s)−µi(s) σi(s) )}} if ξi (s) = 0 ; (2) and for all site s, the parameters {µi,t,s ∈ R}, {σi,t,s > 0} and {ξi,t,s ∈ R} are referred to as the location, the scale and the shape parameters respectively. Particularly, the different values of ξi (s) ∈ R allows (2) to be a spatial EV model, that is, to belong either to Fréchet family, the Weibull one or Gumbel one. The peak-over-threshold approach is common used, fitting data with a generalized Pareto distributions (GPD). This approach, like the coordinatewise one is concerned with asymptotic stochastic behavior of sample of identical copies of variables. In particular, the multivariate GP Distribution of a sample of i.i.d of sequences of variables, is closely linked the underlying MEV one (see Tajvidi (2006)). More precisely, ifH defined the MEV model, then the associated multivariate GP distribution G is defined for all x = (x1, ..., xn) ∈ Rn and for a given x0 = ( x (1) 0 ; ...;x (n) 0 ) in the support of G, H (x1, ..., xn) = { −1 logG(x0) } log [ G(x0 + x) G(min(x, x0)) ] . (3) The major contribution of this paper is to propose new model of stochastic dependence for max-stable processes in spatial and temporal framework. Specifically, section 2 gives the preliminaries of the study. Section 3 deals a new characterization of asymptotic models of D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1037 time-varying models of dependence of spatial processes. In section 4 the analytical forms these spatio-temporal models of dependence for the main usual extremal distributions both for spatial and time varying contexts. 2. Preliminaries This section summaries definitions and properties on the generalized Pareto processes and the copulas of multivariate joint processes dependence which turn out to be necessary for our approach. For this purpose the definition of multivariate copula is necessary. Definition 1. A n-dimensional copula is a non-negative function Cn defined on Rn sat- isfying the following properties. i) Cn(u1, ..., ui−1, 0, ui+1, ..., un) = 0; for all (u1, ..., ui−1, ui+1, ..., un) ∈ In−1. ii) Cn(u1, ..., ui−1, 1, ui+1, ..., un) = Cn−1(u1, ..., ui−1, ui+1, ..., un), that is, an (n-1) copula for all i. iii) The volume VB of any rectangle B = [a, b] ⊆ [0, 1]n is positive, that is, VB = ∑ ε=(ε1,...,εk)∈{0,1}k (−1)s(ε)1+...+in Cn (b1 + ε1 (a1 − b1) , ...bk + εk (ak − bk)) ≥ 0. (4) where a = (a1, ..., an) and b = (b1, ..., bn) . The use of copulas in stochastic analysis whas justified by the canonical parametriza- tion of Sklar, see Joe [9] or Nelsen [12], such that the n-dimensional copula C associated to a random vector (X1, ..., Xn) with cumulative distribution F and with continuous marginal F1, ..., Fn is given, for (u1, ..., un) ∈ [0, 1]n by C(u1, ..., un) = F [F−1 1 (u1), ..., F−1 n (un)]; (5) F−1 being the generalized inverse such as F−1 (x) = inf {t ∈ [0, 1] , F (t) ≤ x}. Even in spatial analysis, stochastic phenomenas can be modeled via copulas. Particu- larly in a spatial context, Schmitz [14] showed that a collection of copulas and marginal distributions also define a stochastic process. So, the above property ii) is given such as, for all collection {Ct1,...,tn ; t1 < ... < tn, n ∈ N} of copulas satisfying the consistent condition lim uk→1− Ct 1,...,t n (u1, ..., un) = Ct 1,...,t n (u1, ..., uk−1, uk+1, ..., un) ; there exists a probability space (Ω, , P ) and a stochastic processes {(Yx) , x ∈ T} such that P (Yt1 < x1, ..., Ytn < xn) = Ct 1,...,t n (Ft1 (x1) , ..., Ftn (xn)) ; (6) and {(Yt) , t ∈ T} is measurable for all t ∈ T. While studying conditional dependence of GPD models, Ferreira et al. (see [7]) have proposed the GP processes as follows. Let C+(S) be the space of non-negative real con- tinuous functions equipped with the supremium norm where S is compact subset of Rd. D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1038 Theorem 1. A stochastic process W is a generalized Pareto process if the following state- ment are satisfied. (a) The expectation E (W (s)/ supu ∈ S) is positive for all s ∈ S, (b) P ( sups∈SW (s)/w0 > x) = x−1 ) for x > 1 (standard Pareto distribution), (c) For all r > w0 and B ∈ B ( C̄+ (S) ) P ( w0W sups∈SW (s) ∈ B ∣∣∣∣ sup s∈S W (s) > r ) = P ( w0W sups∈SW (s) ∈ B ) ; (7) where C+ w0 (S) = { f ∈ C+ (S) : sup f s∈S (s) = w0 } . . In the relation (7) the probability ρ (B) = P ( w0W sups∈SW (s) ∈ B ) is refered as the spectral measure. In a discrete set for S, S = {s1, ..., sn} if W = (W1, ...,Wn) it provides instead : ρ (B) = P w0 (W1, ...,Wn) max 1 ≤ i ≤ n (Wi) ∈ B  . 3. Asymtotic Dependence for Spatio-temporal Processes Even in spatial stochastic context, three possible distributions can describe the asymp- totic behavior of conveniently normalized extremal distributions at a given geographical locality s. These distributions are instead described by a class of dependence models. Specially in a spatial framework, let DN = {s1, ..., sN} ⊂ R2 be the set of locations (ge- ographical ereas, mines localities, ...), sampled over a [ 0, 1 m ] × [ 0, 1 m ] rectangle (m ∈ N), where the phenomenas are observed. Let Y a variable of interest, observed at given site s and date t. Let consider the following notation of component-wise vector of spatio-temporal pro- cess. Y š t (s) = Y (t, s) = {(Yt1 (s1) ; ...;Ytn (sn)) , s ∈ S, t ∈ T} is the response vector at a given time t from a spatio-temporal and max-stable model. So, under this notation a realisation y (t, s) = yt (s) of Yt (s) is obtained as yi,t (s) = µi,t (si) + σi,t (si) ξi,t (si) [ st (s)ξ (i) t (s) − 1 ] for i = 1, ...,m. (8) Equivalently, it comes that, for a given site s DN = {s1, ..., sN} ⊂ R2 P ( Y1(s)−b(1) n (s) a (1) n (s) ≤ y1 (s) ; ...; YN (s)−b(N) n (sn) a (N) n (sn) ≤ yN (s) )n = H (y1 (s) ; ...; yN (s)) . (9) D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1039 For simplicity reasons, let denote, like in the paper [6] that Y (t, s) = Y š t (which is different from Y s t , the s-th power of Yt). Then, under this notational assumption the spatialized version of the joint distribution function F of Y is given by F št for given vector of realization yšt = ( y (1) t (s) , ..., y (m) t (s) ) such as F št (y1 (t, s) , ..., ym (t, s)) = F ( yš11 (t) ; ...; ysmm (t) ) = F (y1 (t, s) ; ...; ym (t, s)) . In the same vein, the spatio-temporal copula associated to the distribution G via Sklar parametrization (1) will be denoted as C št = ( C š1,t; ...;C š m,t ) . So, the relation (5) provides, for all xt = ( x (1) t ; ...;x (m) t ) in Rm × T the relation C št (u1; ...;um) = F št [ š t ( F š1t (u1) )−1 ; ...; ( F šmt (um) )−1 ] . (10) Note that, for all m ∈ N and for all geographical locality s, the spatio-temporal unit simplex of R(m−1 is given, under the notational by ∆š t,m = { λšt = ( λš11 ; ...;λšmt ) ∈ Rm+ ; ∥∥λšt∥∥ =m i=1 λ ši t = 1 } . (11) The following theorem provides an other characterization of the spatio-temporal ex- treme values distribution associated the process {Ys; s ∈ S} . It is a spatio-temporal pa- rameters version of a key result of extreme values theory, see Resnick [13] or Beirlant[1] . Theorem 2. Let { Y š t , s ∈ S, t ∈ T } be a spatio-temporal (ST) process with parametric joint distribution H š t = (H š1 t ; ...;H šm t ). The following statements are satisfied (a) A sufficient condition for the process H š t to be a ST-MEV distribution is that there exists two spatio-temporal non-random sequences { αšn (t) > 0 } and { βšn (t) ∈ R } such that lim n ↑∞ P ( M š t − βšn (t) αšn (t) ≤ yšt ) = ( H1 ( yš1t ) , ...,Hm ( yšmt )) . where M š(i) t is univariate margins of the spatio-temporal componentwise vector of max- ima. (b) Under the condition (a) there exists a ST vector of coefficient λs (t) and ST- dependence function Bš t mapping ∆š t,m−1×S to [ 1 m−1 , 1 ] such that, for all yšt = ( y š(1) t , ..., y š(m) t ) ∈ [0, 1]m, H š t ( yš1 (t) ; ...; yšm (t) ) = exp [ − m∑ i=1 yši (t)Bš t [ λš1 (t) , ..., , λšm (t) ]] ; (12) where { λši ; 1 ≤ i ≤ m } are spatial coefficients. D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1040 Proof. (a) Let {αn > 0} and {βn ∈ R} be the non-random normalizing sequences of H. Then, their corresponding space and time extensions { αšn (t) > 0 } and { βšn (t) ∈ R } are defined on the set, N∗ × S × T, such that lim n →∞ P ( M š t − βšn (t) αšn (t) ≤ yšt ) = lim n →∞ P [ n i=1 ( M ši t − β ši i (t) αšii (t) ≤ yšit )] Then, lim n →∞ P ( M š t − βšn (t) αšn (t) ≤ yšt ) = lim n →∞ P [ m i=1 ( Y ši t ≤ αši (t) yši (t) + βši (t) )] . That is equivalent, due to independence, to lim n →∞ P ( M š t − βšn (t) αšn (t) ≤ yšt ) = lim n →∞ ( m Π i=1 P [( X š i ≤ αši (t) yši (t) + βši (t) )]) . So, there exists a max-stable distribution G whose max-domain of attraction contains the MEV H. Then, lim n →∞ P ( M š t − βšn (t) αšn (t) ≤ yšt ) = lim n →∞ [ G ( αšiyi š (t) + βši (t) ) , ...αši (t) yš (t) + βši ]n . Finally, since the distribution G is max-stable lim n →∞ P ( M š t − βšn (t) αšn (t) ≤ yšt ) = ( H1 ( y š(1) t ) , ...,Hn ( y š(n) t )) . (b) Assume that the distribution H is a MEV model, that is its univariable marginal Hi satisfies relation (9). Therefore, it is sufficient to show for a given site s and date t, that, H š t satisfies the spatio-temporal version of max-stability property. It comes from Coles ([7]) that, at a given site s and date t the MEV model H has the following representation Yi such as yi = −1 log [1− λiti(si)] with si > ui. Note moreover that it not be restrictive to assume in the following that the spatio- temporal multivariate process { Y š k (t) , s ∈ S, t ∈ T } has spatio-temporal unit Fréchet mar- gin, which it is more convenient to work with. Y š t ∼ Φš θ,t ⇔ ln ( Y š t )θ ∼ Λšt ⇔ −1 Y št ∼ Ψš θ,t ⇔ Y š t = µ ( yšt ) + σ(yšt ) ξ(yšt ) [( yšt )ξ(yt) − 1 ] . Therefore, H (y1, ..., ym) = exp [−Sm max (q1λ1t1(s1) , ..., qmλmtm(sm))µd(q)] + o(max(λi). If, in particularly, for all i = 1, ..., n we set λiti(si) = λt (s), then it follows that there exists a spatio-temporal dependence function Bš t = B(λ, q, y) such as:, Bš t (λ) = λ i=m∑ i=1 ti(xi)Sm max ( q1t1(x1)∑i=m i=1 ti(xi) , ..., qm ( 1− ∑i=m−1 i=1 ti(xi)∑i=m i=1 ti(xi) )) µd(q). D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1041 Particularly under the above component-wise notation H š t (yš1 (t) , ..., yšn (t)) = exp [ − ( m∑ i=1 yši (t) ) Bš t ( −q11y š 1 (t)∑m i=1 y š i (t) ; ...; −qm−1y š m−1 (t)∑m i=1 y š i (t) )] . (13) Moreover, taking into account Dossou et al., it follows that Bš t ( −q11y š 1 (t)∑m i=1 y š i (t) ; ...; −qm−1y š m−1 (t)∑m i=1 y š i (t) ) = D ( −q11yš1(t)∑m i=1 y š(i) t , . . . , −qm−11y š(m−1) t∑m i=1 y š(i) t ) + ( 1- −q11yš1(t)∑m i=1 y š(i) t ) DN̄1  −q11yš1(t)∑m i=1 y š(i) t 1− −q11yš1(t)∑m i=1 y š(i) t ; . . . ; −qm−11y š(m−1) t∑m i=1 y š(i) t 1− −q11yš1(t)∑m i=1 y š(i) t  . Finally, by noting λsi (t) = −qi∑m i=1 y š i (t) it follows that Bš t ( λs1, ..., λ s m−1 ) = D ( λs1y š 1 (t) , ..., λs1y š 1 (t) ) + (1− t)D ( λs1y š 1 (t) , ..., λs1y š 1 (t) ) where Bš t is the spatialized Pickands dependence function, mapping the simplex ∆s,m−1 to [ 1 m−1 ; 1 ] (see Beirlant [1] . Thus, we obtain the result as asserted Definition 2. The space and time dependent function Bš t ( λs1, ..., λ s m−1 ) is called the Spatio-temporal Asymptotic Dependence (STAD) function associated to the process {Ys}. Particularly, in a the following and with a parameter θ we can set Bš θ,t (λs) = Bš θ (λt) where λt ∈ ∆š t,m. For example, for the bivariate and one parametric negative logistic model (see Joe [9]) defined for yšt = ( y š(1) t , y š(2) t ) and θ = (θ1, θ2) ≥ 0 by Gšθ ( yšt ) = exp { − ( 1 y š(1) t + 1 y š(2) t − [( y −š(1)θ1 t y −š(2)θ1 t ) −θ1 ]−1 θ )} ; then, it follows that the corresponding ST dependence function is given by Bš θ (λst ) = 1 1 + λst [ 1− ( 1 + λs−θt )−1 θ ] with λst ∈ [0, 1] The following theorem, proposes a spatial characterization the multivariate GP distri- bution associated to the spatial MEV of the same process Y. Theorem 3. Let { Gšt ,s ∈ S, t ∈ T } be a MEV distribution of a sample of copies of a spatio-temporal max-stable process Xs t and Hš t the multivariate GP associated to the same sample. Then, for a given site s0 and date t0 , H š t (yst ) = −1 logGšt (yst0 ) log ( Gšt (yst0 +yst ) Gšt ( min(yst ,y s t0 ) ) ) = 1− log ( Gšt (yst ) Gšt ( min(yst ,y s t0 ) ) ) . (14) for all yst0 ∈ support(G š t ). D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1042 Proof. It should be noted that the normalizing sequences {αn > 0} and {βn ∈ R} in Theorem 5 are given by σn = F−1(1− 1 n ) and βn = f (σn ) 1− F (σn ) ; where f is the common density function of the sample. Moreover, it should be considered that in this section operations on vectors are componentwisely, that is, for a given location s of S;  ak (s0) = ( a (1) k (s0,t) ; ...; a (m) k (s0,t) ) ak (s) + bk (s) = ( a (1) k (s) + a (1) k (s) ; ...; a (m) k (s) + a (m) k (s) ) yst yst0 = ( y s(1) t y s(1) t0 ; ...; y s(m) t y s(m) t0 ) (15) Let H be a given multivariate Pareto distribution. So for a given point yst0 = ( ys1t0 , ..., y sm t0 ) with ysit0 > 0 and α > 0, it follows that H š t (yst ) = 1− [ yst y s(1) t0 ]−α . In particular, for y s(1) t > y s(1) t0 , it commes that H−1 (yst ) = (1− yst ) −1/α y s(1) t0 . Therefore, we have: an (s) = G−1 ([ 1− 1 n ]) = n1/αyst0 . Otherwise, asymptotically, it comes that lim n↑+∞ P (Mn (s) ≤ an (s0) yst ) = lim n↑+∞ [ 1− ( n1/αyšt yšt0 )−α]n that is lim n↑+∞ P (Mn (s) ≤ an (s0) yst ) = lim n↑+∞ [ 1− n1/αy−sαt ] which gives marginally lim n−→+∞ P (Mn (si) ≤ an (s0) ysit ) = exp ( −1 y š(i) t ) = H ši t ( yšit ) Furthermore, H š t (yst ) = { 1− log ( G(yst ) G ( min(yst , y s t0 ) )) si yst ≥ 0 0 elsewhere .(16) Finally, using simultanously the relations (4) and (22) we obtain (20) as asserted D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1043 4. Analytical Characterization of STAD Note that it should be noted that even in spatio-temporal context, the dual relation (see [8]) relying the vectors of maxima and minima holds. So, ,{ min { Y š k (t) } 1 ≤ k ≤ m } = − { max { −Y š k (t) } 1 ≤ k ≤ m } for all site s ∈ S and date t ∈ T. Most of these families arise from symmetric, asymmetric or mixed extensions of a known differentiable parametric model: the logistic family (see Degen [4]). 4.1. The Pseudo-Power function of STAD Theorem 4. Let H š θ,t be the parametric and max-stable distribution modeling the stochas- tic behavior of a space and time varyng process. Then there exists a mutivariate parametric pseudo-power function P šθ such as Gšθ ( ỹš1t , ..., ỹ šm t ) = exp { −P šθ ( ỹš1t , ..., ỹ šm t )} , where P šθ is defined on R× S × T. Proof. In the proof of theorem 5, the relation (13) shows that Particularly under the above component-wise notation H š t (ỹš1t , ..., ỹ šm t ) = exp [ − ( m∑ i=1 yšit ) Bš t ( −q1y š1 t∑m i=1 y ši t ; ...; −qm−1y šm−1 t∑m i=1 y ši t )] (17) By setting P šθ ( ỹš1t , ..., ỹ šm t ) = ( m∑ i=1 yšit ) Bš t ( −q1y š1 t∑m i=1 y ši t ; ...; −qm−1y šm−1 t∑m i=1 y ši t ) . On obtain a pseudo power function P šθ in yšit for i = 1, ..., n Remark 1. To characterize a spatio-temporal max-stable model consists simply to provides the underlying pseudo-power function 4.2. Analytical Form of Bivariate STAD This section, we provide the analytical forms of the ST models of the dependence of the main usual families of extreme distributions. According to remark 7, it is sufficient to gives the corresponding pseudo-power function P šθ ( ỹšt ) where ỹšt = ( ỹš1t , ỹ š2 t ) . STAD of Logistic model and symmetric entensions D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1044 1 Logistic model (Gumbel family) with θ ≥ 1 (voir Joe [9]) · P šθ ( ỹšt ) = (( ỹš1t )θ + ( ỹš2t )θ) 1 θ ; · Bš θ(λt) = λt 1 + λt [( 1 + λ−θt ) 1 θ − 1 ] 2 Negative one-parametric logistic model (Galambos family) with θ ≥ 0 · P šθ ( ỹšt ) = ( 1 ỹ š1 t + 1 ỹ š2 t − [ 1 ỹ š1θ1 t + 1 ỹ š1θ1 t ]−1 θ ) ; · Bš θ(λt) = λt 1 + λt [ 1− ( 1 + λ−θt )−1 θ ] 3 Negative two-parametric logistic model or model of Joe (see [9] ); θ = (θ1, θ2) · P šθ ( ỹšt ) = ( yš1t + yš2t − [ y−š1θ1t + y−š2θ1t − ( ỹš1θ1θ2t + yš2θ1θ2t )− 1 θ2 ] 1 θ1 ) · Bš θ(λt) = λt 1 + λt ([ λ−θ1t + 1− ( λθ1θ2t + 1 )−1 θ2 ] 1 θ1 − 1 ) . 4 Gaussian bivariate model (or model of Hüsler-Ré̈ıss) with θ ≥ 0 (see [8]) · P šθ ( ỹšt ) = [ ỹš1t Φ ( 1 θ + θ 2 log ( ỹš1t ỹš2t )) + ỹš1t Φ ( 1 θ + θ 2 log ( ỹš2t ỹš1t ))] Φ being the cumulative distribution function of N(0,1). · Bš θ(λt) = λt 1 + λt [ 1 λt Φ ( 2−θ2 log(λt) 2θ ) − Φ ( −2+θ2 log(λt) 2θ )] . 5 Symmetric extension of logistic model or model of Tajvidi (see [12]) · P šθ ( ỹšt ) = exp { − [(( ỹš1t )θ1 + ( ỹ š(2) t )θ1) + θ2 ( ỹš1t ỹ š2 t ) θ2 2 ] 1 θ1 } ;with θ = (θ1, θ2) · Bš θ(λt) = λt 1 + λt [ λt −θ1 + 1 + θ2λt −θ1 2 ] 1 θ where 0 < θ2 ≤ 2(θ1 − 1); θ2 ≥ 2 6 Symmetric two-parametric extension of logistic model θ = (θ1, θ2) (Joe [9]) ◦ P šθ ( ỹšt ) = [(( ỹš1t )θ1 + ( ỹš2t )θ1)− θ2 (( ỹš1t )θ1θ2 + ( ỹš2t )θ1θ2) 1 θ2 ] 1 θ1 · Bš θ(λt) = λt 1 + λt ([ λt −θ1 + 1− ( λt θ1θ2 + 1 )−1 θ2 ] 1 θ1 ) where θ2 > 0; θ1 ≥ 1 7 Symmetric Extension of bilogistic model, proposed by Smith (see Michel [11]) ◦ P šθ ( ỹšt ) = ( ỹš1t q 1−θ1 + ỹš2t2 (1− q)1−θ2 ) 1 θ1 with θ = (θ1, θ2) ; 0 < θ1 ; θ2 < 1 where q = q(θ1, θ2) are the roots of equation: 1− θ1)ỹ š(1) t (1− q)θ2 − (1− θ2)ỹ š(2) t qθ1 = 0 · Bš θ(λt) = λt 1 + λt [ q1−θ1 + λt(1− q)1−θ2 + 1 ] . While studying max-stable models Joe (see [9]) and Tajvidi (see [15]) have proposed many asymmetric extension of logistic model. STAD of logistic model and asymmetric generalizations D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1045 1 Asymmetric three parametric extension of logistic model with θ = (θ1, θ2, θ3) · P šθ ( ỹšt ) = (1− θ2) ỹš1t − (1− θ1) ỹš2t − [( θ1ỹ š1 t )θ3 + ( θ2ỹ š2 t )θ3] 1 θ3 ; · Bš θ(λt) = λt 1 + λt [ 1− θ1 + θ2λt+ ( θθ31 + (θ2λt) θ3 )] 1 θ3 with θ1 ≥ 0, θ2 ≤ 1, θ3 ≥ 1 2 Asymmetric three parametric and negative extension of logistic model · P šθ ( ỹšt ) = ( ỹš1t + ỹš2t ) + [( θ1ỹ š1 t )−θ3 + ( θ2ỹ š2 t )−θ3]−1 θ3 with θ = (θ1, θ2, θ3) · Bš θ(λt) = λt 1 + λt [ 1− ( θ−θ31 + (θ2λt) −θ3 )]−1 θ3 where 0 < θ1, θ2 ≤ 1, θ3 > 0 3 Symmetric one parametric, mixed extension (proposed by Tajvidi (see [11]) · P šθ ( ỹšt ) = [( ỹš1t + ỹš2t ) − θ2 ( ỹš1θ1 + ỹš2t θ1 ) 1 θ1 ] with θ ≥ 0 · Bš θ(λt) = 1 1 + λt [ 1− θ2 [ 1 + λ−θ1t ]− 1 θ1 ] . 4 Asymmetric two-parametric model (proposed by Coles and Tawn ([10]) · P šθ ( ỹšt ) = [( ỹš1t + ỹš2t ) + [1−B(q, θ1 + 1, θ2)] ỹš1t + ỹš2t B(q, θ1, 1 + θ2) ]−1 θ3 where B(q, θ1, θ2) is Beta distribution at q(θ1, θ2) = θ1ỹ š1 t θ1ỹ š1 t + θ2ỹ š2 t . · Bš θ(λt) = λt 1 + λt [ (1−B(q, θ1 + 1, θ2)) λt +B(q, θ1, 1 + θ2)− 1 ] with θ1, θ2, θ3 > 0. 5 Bilogistic and negative model ( proposed by Müler (see Joe [9]); · P šθ ( ỹšt ) = ( ỹš1t + ỹš2t ) − ỹš1t q1+θ1 + ỹš2t (1− q)1+θ2 where q = q(ỹš1t , ỹ š2 t , θ) is root of equation with θ = (θ1, θ2) > 0 (1 + θ1)ỹš1t q θ1 − (1 + θ2)ỹš2t (1− q)θ2 = 0 · Bš θ(λt) = λt 1 + λt [ 1− q1+θ1 − λt(1− q)1+θ2 ] 6 Symmetric mixed polynomial model of Klüppelberg (see Beirlant [1]) · P šθ ( ỹšt ) = − ( ỹš1t + ỹš2t ) + θ ỹš1t + θỹš1t ỹš1t + ỹš2t · Bš θ(λt) = 1 1 + λt [ (1− θ) + θ 1 + λt ] where θ ∈ [0, 1] 7 Asymmetric, mixed polynomial model of Klüppelberg (see Beirlant [1]) · P šθ ( ỹšt ) = ( ỹš1t + ỹš2t ) − (θ1 + θ2) ỹš1t − θ1ỹ š1 t ỹš1t + ỹš2t − θ2ỹ š2 t( ỹš1t + ỹš2t )2 with θ = (θ1, θ2, θ3) where: θ1 ≥ 0, θ1 + 3θ2 ≥ 0; θ1 + θ2 ≤ 1; θ1 + 2θ2 ≤ 1 · Bš θ(λt) = 1 1 + λt [ (θ1 + θ2) + θ1 1 + λt + θ2 (1 + λt) 2 ] . D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1046 4.3. Analytical Form of Tridimensional STAD We provide analytical form of the STAD function of three dimensional logistic model (see [5] and [4]). In this sub-section, let consider ỹšt = ( ỹš1t , ỹ š2 t , ỹ š3 t ) and λt = ( λ (1) t , λ (2) t ) D. Barro, S. P. Nitiéma, M. Diallo / Eur. J. Pure Appl. Math, 10 (5) (2017), 1035-1049 1047 1 Trivariate logistic model of ST- max-stable distribution (see [9]) · Pθ(yšt ) = [ ỹš1θt + ỹš2θt + ỹš3θt ] 1 θ where θ ≥ 1. · Bš θ(λt) = [ λ (1) t θ + λ (2)θ t + ( 1− λ(1) t − λ (2) t )θ]1 θ − [ λ (2) t θ + ( 1− λ(1) t − λ (2) t )θ]1 θ 2 Negative, two parametric extension of trivariate logistic model (see [8]) · Pθ(yšt ) = − ([ ỹš1θ1θ2t + 2−θ2 ỹš2t θ1θ2 ] 1 θ2 + [ 2−θ2 ỹš2t θ1θ2 + ỹš3t θ1θ2 ] 1 θ2 ) 1 θ1 · Bš θ(λt) = [λ(1) t θ1θ2 + 2−θ2λ (2) t θ1θ2 ] 1 θ2 + [ 2−θ22 λ (2) t θ1θ2 + ( 1− λ(1) t − λ (2) t )θ1θ2 3 ] 1 θ2  1 θ1 − [ 2−θ2λ (2) t + ( 1− λ(1) t − λ (2) t )−θ1θ2 1 θ2 ] 3 Trivariate Gaussian model of ST- max-stable distribution (see [8]) · Pθ(yšt ) = ∑3 i=1 ỹ ši t + 1 ỹ š1 t [ Φ ( 1 θ1 + θ1 2 log ( ỹ š2 t ỹ š1 t )) + Φ ( 1 θ3 + θ3 2 log ( ỹ š3 t ỹ š1 t ))] − 1 ỹš2t [ Φ ( 1 θ1 + θ1 2 log ( ỹš1t ỹš2t )) + Φ ( 1 θ2 + θ2 2 log ( ỹš3t ỹš2t ))] − 1 ỹ š3 t [ Φ ( 1 θ3 + θ3 2 log ( ỹ š1 t ỹ š3 t )) + Φ ( 1 θ2 + θ2 2 log ( ỹ š2 t ỹ š3 t ))] + I ( Φ̄2, θ, ρ ) ; I being the integral I = y−1 3 0 Φ̄2 ( 1 θ2 + θ2 2 log ( ỹ š(1) t ) ; 1 θ3 + θ3 2 log ( ỹ š(4) t ) ; ρ ) dt and ρ =  0 2 ( 1 θ2 1 + 1 θ2 2 + 1 θ3 3 ) 2 ( 1 θ2 1 + 1 θ2 2 + 1 θ3 3 ) 0  ;covariance-matrix. · Bš θ(λt) = 2− λ(1) t [ Φ ( 1 θ1 + θ1 2 log ( λ (1) t λ (2) t )) + Φ ( 1 θ3 + θ3 2 log ( λ (1) t 1−λ(1) t −λ (2) t ))] −λ(2) t [ Φ ( 1 θ1 + θ1 2 log ( λ (1) t λ (2) t )) + Φ ( 1 θ2 + θ2 2 log ( λ (1) t 1− λ(1) t − λ (2) t ))] − ( 1− λ(1) t − λ (2) t )[ Φ ( 1 θ3 + θ3 2 log ( 1−λ(1) t −λ (2) t λ (1) t )) + Φ ( 1 θ2 + θ2 2 log ( 1−λ(1) t −λ (2) t λ (1) t ))] −λ(2) t Φ ( 1 θ2 + θ2 2 log ( λ (2) t 1−λ(2) t )) − ( 1− λ(2) t ) Φ ( 1 θ2 + θ2 2 log ( 1−λ(2) t λ (2) t )) +R(λ (1) t , λ (2) t , θ). 4 An asymmetric extension of logistic ST- max-stable distribution (see [9]) · P šθ ( ỹšt ) = 3∑ i=1 ỹšit − ( ỹ−š1θ1t + ỹ−š3θ2t ) )−1 θ1 + [ ỹš1θ1 t + ỹš2θ1 t − ( ỹš2θ1θ2t + 2θ1 ỹ š(2)θ1θ2 t )−1 θ2 ]−1 θ1 + [ ỹš2θ1 t + ỹš3θ1 t − ( ỹš3θ1θ2t + 2θ1 ỹš2θ1θ2t )−1 θ2 ]−1 θ1 − [ −ỹ−š1θ1 t + ỹ−š2θ1 t + ỹ−š3θ1 t − ( ỹš1θ1θ2t + 2θ1 ỹš2θ1θ2t )−1 θ2 − ( ỹš3θ1θ2t + 2θ1 ỹš2θ1θ2t )−1 θ2 ]−1 θ1 . · Bš θ(λt) = λ (1) t − ( λ (1) t θ1 + ( 1− λ(1) t − λ (2) t )θ1)−1 θ1 + [ λ (1)θ1 t − ( λ (1)θ1θ2 t + 2λ (1)θ1θ2 2 ) −1 θ1 ]−1 θ2 + [( 1− λ(1) t − λ (2) t ) + λ −(1)θ1 t + λ −(2)θ2 t − ( λ −(1)θ1θ2 t + 2λ −(2)θ1θ2 2 ) −1 θ1 ]−1 θ2 . 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