Extremal Dependence Modeling with Spatial and Survival Distributions

This paper investigates some properties of dependence of extreme values distributions both in survival and spatial context. Specifically, we prospose a spatial Extremal dependence coefficient for survival distributions. Madogram is characterized in bivariate case and multivariate survival function and the underlying hazard distributions are given in a risky context.


Introduction
Extreme values (EV) analysis finds wide applications in many areas including climatology, environment sciences (Beirlant, J., et al., 2005), risk management (Balkema, G. & Paul, E., 2007;Degen, M. & Embrechts, P., 2008) and survival analysis (Hougaard, P., 2000).The distributions of this domain can be obtained as limiting distributions of properly normalized maxima of independent and identically distributed random variables.In particular if is a max stable random field defined on a set X = {x 1 , ..., x k } , then the spatial EV analysis shows that Z results from observations of a stochasltic process such as ]} with s ∈ D; (1.1) provided the limit exists, where {a n (.) > 0; n ≥ 1} and {b n (.) , ; n ≥ 1} are sequences of real constants, s being a spatial location of a domain D ⊂ R d and Z (s), a random quantity (Padoan, S. A., et al., 2010).
Survival analysis is a subdomain of statistics which deals with failure or death time or natural catastroph.It is a important topic in many areas including biomedical, biostatistics, environment, etc (Padoan, S. A., et al., 2010;Resnick, S. I., 2008).One may distinguish three kind of models in survival analysis: the non parametric models, the semi-parametric models and the parametric ones.
Let T = (T 1 , ..., T n ) be a vector of lifetimes of n individuals in a given population with distribution F T .If in particular T describes the life long time, the fraction of the population which will survive past a given vector of times t = (t 1 , ..., t n ) is provided by the survival distribution, conventionally denoted S T , such as S T (t 1 , ..., t n ) = FT (t 1 , ..., t n ) = P(T 1 ≥ t 1 , ..., T n ≥ t n ). (1. 2) The hazard function h T of T specifies the instantaneous rate of failure (risk or mortality rate) at a given date t given that the individual survived up to time t.If the margins are absolutly continuous the cumulative density function (cdf) is also related to S T such as Spatial analysis is a key component of statistic involving a collected from different locations.In particular, while studying in biostatistics, epidemiology, environment sciences, data have a common, that they are collected from different spatial locations and they are nether independent nor identically distributed.So, that in spatial framework, when survival times are spatially referenced, some of clusters of high or low times might be apparent on a visual inspection of the data.The question which naturally arises as to whether these observed spatial survival patterns can be explained by incorporating appropriate covariates into the model or whether, in order the unexplained spatial variation.
The main contribution of this paper is to investigate some asymptotic properties of multivariate dependence models both in survival and spatial context.Section 2 deals with spatial measures of extremal dependence.In particular the extremal dependance spatial coefficient is modeled and survival madogram is characterized in bivariate case.In Section 3, the survival and hazard distributions are given in a risky context.

Spatial Extremal Dependence Coefficient
In multivariate extreme values (EV) analysis, many related measures have been proposed for quantifying the magnitude of the extremal dependence when the random vector exhibits asymptotic dependence.In particular, in univariate EV study, even in spatial and survival framework, three types of distributions can summary the asymptotic behavior of conveniently normalized maximum of distributions (Beirlant, J., et al., 2005).
For a fixed k in ) denote independent copies of stochastic process observed at given ereas s of a domain S. Assume that the process {Y (s) , s ∈ S } is parametric max-stable distribution.Then the asymptotic distribution modeling the stochastic behavior is the same type like one of the three extremal spatial distributions . (2.1) Let T s = (T 1 (s) , ..., T n (s)) be a vector of lifetimes of n individuals in a given population observed at a given site s of spatial domain S = { (s 1 , ...., s m ) , s j ∈ R 2

}
. The process T s is the survival and stochastic random vector which with joint distribution F s = ( F s,1 ; ...; F s,n ) .Therefore, for all realization y, In all this study, our key assumption is that the process T s is continuous, stationary and is max-stable with generalized Fréchet margins.So, for a given site s in S ξ where u + = max (u, 0) and Notice that such an assumption implies no loss of generality since even in survival and space-varying context, every onedimensional EV distribution can be obtained by a functional transformation of another.In particular, if for a given site s, Among measures of extremal dependence there are the extremal coefficient (Hougaard, P., 2000) or the madogram and its nested model the link between two sets of R d (Cooley, D., et al., 2006).Moreover and for simplicity reason let's denote like in (Barro, D., et al., 2016) that: ) . Under the restriction to the simplest case where F i )) = 0 if i j, the following result allows us to provide a characterisation of the spatial extremal dependence (SED) in a survival field.
Theorem 1 Let T s be a vector of lifetimes of n individuals in a given population with distribution Fs satisfying the key assumption.
i) The one-dimensional marginal law ii) There exists spatio-survival parametric measure of probability P ξ(s) defined on R 2 × {s} and a non-decreasing function g s defined on D ξ such that the survival and spatial extremal coefficent θ s ) of the process is given by where h i j = s i − s j is the separating distance between these sites s i and s j .
Proof.By assumption the distribution function T s of the process is max-stable.So, for all site s, there exist vectors of constants where G i is an EV distribution and M i (x i ) is the spatial survival vector of the maximum But since F i is the marginal distribution of T s then, it lies on the max-domain of attraction of G i .Thus, for all site s i , the relation (2.5) is equivalent to the generalized EV model, given by lim where {σ i (s i ) > 0} and {µ i (s i ) ∈ R} and {ξ i (x t ) ∈ R} such that, for all i,1 ≤ i ≤ n are respectively the spatio-survival parameters of location, scale and shape of the observation at the parametric site s i .
ii) In bivariate case and for all pair of sites s i and s j the extremal dependence parametric coefficient θ depends on the of separating distance h i j .
It follows that (2.6)Moreover, using in the relation (2.6) the general form of a univariate EV model with normalizing coefficients σ > 0, µ ∈ R, ξ i (s i ) ∈ R, it comes, in the particular bivariate context, that Then, by introducing the concept of probability measure the relation (2.7) is equivalent to = P s i j ,ξ s and finally Thus we obtain (2.4) as asserted The following proposition provides a consequence of theorem 1 Corollary 2 Let {T s ; s ∈ S } a spatial process satisfying the key assumption.Then, f or all site s i ∈ S i) the marginal survival parametric extremal density f ξ i is given by ii) the parametric hazard functions h ξ (t) are given (2.9) Proof.In such a case, the parametric survival function is given marginally by (2.10)

Distortional Function of Spatial Extremal Model
The following result characterizes a multivariate survival distribution via a spatial and distortional measure of dependence.
Proposition 3 Let {T s ; s ∈ S } a spatial process satisfying the key assumption.Then there exists a spatial conditional dependence measure such as D s defined on the spatial unit simplex, for all xs = (2.12) Proof.The EV analysis results from asymptotic normalized vector of maxima of a random vector which converges to a non degenerated multivariate EV model G.One of extremal study approach is the Peacks-over threshold (POT).
where ÃF is a spatio-survival dependence function of Pickands associated to F .
Furthermore, for a given 1 ≺ N ≺ n let consider the N-partition of the spatial domain S proposed in [9] Then, it follows that the corresponding distorsional probability δs is such that; ) .

Moreover let Fs
N and Fs N be the corresponding partitional distributions functions.So, it comes that ) .
Furthermore, from results due to Dossou et al. (Dossou,-G. S., 2009) Which can be written equivalently, where Ds being a distortional spatial and survival dependence function ) Particularly in bivariate case it is easy to show that ] by .
Specially, for the logistic model: which is given graphically has follows Figure 1.Bivariate logistic model for θ 1 = θ 2 = 2

A New Charaterization of Survival Madogram
Madogram is a measure of the full pairwise extremal dependence function to evaluate dependence among extreme regional observation.Some extensions of this tool have been proposed.Specifially while modeling spatial extreme variablity of an isotropic and max-stable field, Cooley (Cooley, D., et al., 2006) proposed the F-madogram γ F (h) which transforms the process via its marginal F. The following result provides a parametrization of characterizing of the madogram.
Proposition 4 Let {T s ; s ∈ S } a spatial process satisfying the key assumption with distribution Fs .Then, the survival λ-madogram associated to the bivariate margins of F is given by the ratio where P and Q are convenient polynoms and D h being a distortional spatial dependence measure.
Proof.In the previous proposition, the bivariate case implies that, particularly in bivariate case it is easy to shows that ] by .
Furthermore, in condional study, Proposition 6 of the paper (Barro, D. et al., 2012) provides that, under additional contraints, the λ−madogram can be expressed as where D h is a conditional spatial measure convex defined on the unit simplex of R 2 .
In our context by replacing D(t ( xs )) by D(t ( xs )) it follows easily that where D h is a conditional spatial measure convex defined on the unit simplex of R2 In epidemiological studies, the intensity of contamination must change over time.For example, at begining of the epidemy, the intensity is high but it decreases when the sanitaries autorities take some dispositions epidemy.
Let T s = (T 1 (s) , ..., T n (s)) be the survival, continuous and stochastic random vector T satisfying the key assumption.
Consider H as a closed half space in R n where P (T s ∈ H) > 0. As in the high scenarios defined by Degen (see [1], [5] and [6]), the following results characterize the spatial and survival risk and some derivative properties.
Definition The spatial and survival high risk scenario T H s associated to the process T s is defined as the vector T(s) conditioned to lie in the half space H.The spatial probability distribution function (pd where, ts,i = FH i,s (t i ) are the marginals distributions functions of the distribution T H and given Proof.Let (X 1 , ..., X n ) , n ∈ N, be a vector of random i.i.d variables with a joint distribution F with continuous margins F i .
According to Sklar's theorem (see [15]), there exists a unique copula, C T providing a canonical parameterisation of F via its univariate marginal quantile functions F −1 i such that, Or conversively, for C F providing a canonical parameterisation of F via its univariate marginal quantile functions Differentiating the formula (3.4) shows that the density function of the copula is equal to the ratio of the joint density h of H to the product of marginal densities h i such as, for all (u 1 , ..., ) Thus, we obtain the relation (3.3) as asserted

Conclusion
The results of this study show that the survial and spatial framework are also convenient to model extremal dependence.Tools of dependence such as the extremal dependance coefficient, the multivariate dependence function, the madogram have been modeled both in spatial and survival context.The survival and hazard distributions are given in a risky context.

Figure 2 .
Figure 2. Bivariate distortional λ − madogram Then the vector of exceedances of the same sample have a generalized Pareto model H. Particularly if the extremal function Fs underlying the survival process T S .It follows that its spatio-survival associated POT model Hs satisfies, for all xs = n)∈ Rn , the relationship by both the partitional distributions functions Fs N and Fs N lie also in the max-domain of attraction of two multivariate EV distribtutions.And by noting ÃN and Ã N the Pickands dependence functions of t Fs then Z H s has the high risk distribution π H given by Proposition If T s has the distribution Fs , then T H s has the high risk distribution FH s given by, , for all i = 1, . . ., m.