Estimating the support of a Poisson process via the Faber-Schauder basis and extreme values

Let N be a Poisson process of unknown intensity c and of support S where S is given by {(x,y) ∈ R 2 | 0 ≤ x ≤ 1 and 0 < y < f(x)}. The purpose of this paper is to determine an estimator of the support S, i.e. an estimator of the edge function f, given the observed points of the Poisson process N. By estimating with a linear combination of extreme values the coefficients of a truncated Faber-Schauder series, we obtain an estimation of f. We establish different kinds of convergence of the estimator and give conditions for two different possible limit distributions. Finally, we present and illustrate through a simulation a simple bias correction of the estimator.