How sharp are classical approximations for statistical applications?

This paper aims at comparing theoretical approximations of the tail of the maximum of stochastic processes and the corresponding numerical evaluations. More particularly, we focus on the Pickands or double sum method, the Rice method, the Euler Characteristic method and a new one called the Poisson method. The numerical evaluation, performed using mainly Quasi Monte-Carlo integration and adaptations of the programs of Genz, show the domains of validity of each method