International Journal of Statistics and Probability; Vol. 1, No. 2; 2012 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education Spatial Stochastic Framework for Sampling Time Parametric Max-stable Processes Barro Diakarya 1 , Blami Kot´ e 2 & Souma¨ ıla Moussa 2 1 UFR-SEG, Universit´ e Ouaga 2, Laboratoire Lanibio 2 UFR-SEA, Universit´ e de Ouagadougou, Laboratoire Lanibio, Burkina Faso Correspondence: Diakarya Barro, Universit´ e Ouaga 2, UFR-SEG, 09 BP: 1654 Ouagadougou 09, Burkina Faso. Tel: 226-78-601-0686. E-mail: dbarro2@gmail.com Received: September 2, 2012 Accepted: September 24, 2012 Online Published: October 17, 2012 doi:10.5539/ijsp.v1n2p203 URL: http://dx.doi.org/10.5539/ijsp.v1n2p203 Abstract Modelling the spatial extreme events uses the approach of max-stable processes which describe the stochastic behaviour of point-referenced data. Max-stable processes form the natural extension of multivariate extreme values distributions to infinite dimensions. In this paper we consider a max-stable stochastic process over space index. We extend the modelling to time-varying setting using new characterizations of the multivariate distribution underlying the process. A distortional measure is introduced to describe the marginal laws and joint dependence. Keywords: spatial, max-stable, stochastic process, extreme values distributions 2000 Mathematics Subject Classification: 60G52, 60G70, 62G20, 62G32 1. Introduction Stochastic processes play a fundamental role in modelling data behaviour on a set of spatial observations. Max- stable processes are stochastic distributions over some index set (space or time) such that all the finite dimensional distributions are max-stable. Space-time (ST) max-stable processes form a class of asymptotically justified models describing spatial dependence among extreme time valued. Max-stability is the foundation of multivariate extreme values (MEV) analysis. With a vast field of applications, statistics of extremes are concerned with modelling asymptotic behaviour of distributions mainly for component-wise maxima of laws when appropriatetly normalized. In one-dimensional extreme setting the well-known theorem of Fisher-Tippet-Gnedenko (See Beirlant) shows that three possible distributions can characterize these asymptotic behaviours such as: Λ( x) = exp  - exp (-x)  ; ∞ < x < ∞, Gumbel distribution Φ θ ( x) = exp  -x -θ  ; x > 0,θ> 0, Fr´ echet distribution Ψ θ ( x) = exp  - (-x) θ  ; x > 0,θ> 0, Weibul distribution ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (1) Multivariate extension of the univariate case leads instead to various non-trivial problems and no fully parametric model can summary the whole families. In a spatial framework, modelling of extremes is an important adequat risk management in environment sciences. Indeed, many environment extremal problems are spatial or temporal in extent (sea height, annual maxima, daily rainfall, snow depth etc.). Specifically, the prospect of climatic change and its impact have brought spatial statis- tics of extreme events into sharper focus. Compared to their applications to classical statistics, the use of extreme values settings to spatial analysis is still in an evolutionary stage. In a pioneering work in this field (Smith, 1990; unpublished data) proposed a approach to model continuous max-stable processes using the canonical represen- tations of de Haan (see Coles, 2001) of the underlying MEV distributions. This approach have been applied to ozone data in North Carolina by Naveau et al. (2006), to rainfall data by Smith and Stephenson (2009), Padoan et al. (2010) and Ribatet et al. (2010). More recently Blanchet and Davidson (2011) used max-stable processes to model maxima of annual snow height. All these approaches for modelling spatial aspects of rare events are based on generalization of the classical MEV distributions whose margins are usually standardized to unit-Fr´ echet models (Beirlant et al., 2005). 203