  Parametric inference for spatial max-stable processes is difficult since the
related likelihoods are unavailable. A composite likelihood approach based on
the bivariate distribution of block maxima has been recently proposed in the
literature. However modeling block maxima is a wasteful approach provided that
other information is available. Moreover an approach based on block, typically
annual, maxima is unable to take into account the fact that maxima occur or not
simultaneously. If time series of, say, daily data are available, then
estimation procedures based on exceedances of a high threshold could mitigate
such problems. In this paper we focus on two approaches for composing
likelihoods based on pairs of exceedances. The first one comes from the tail
approximation for bivariate distribution proposed by Ledford and Tawn (1996)
when both pairs of observations exceed the fixed threshold. The second one uses
the bivariate extension (Rootzen and Tajvidi, 2006) of the generalized Pareto
distribution which allows to model exceedances when at least one of the
components is over the threshold. The two approaches are compared through a
simulation study according to different degrees of spatial dependency. Results
show that both the strength of the spatial dependencies and the threshold
choice play a fundamental role in determining which is the best estimating
procedure.
