Functional estimation and hypothesis testing in nonparametric boundary models

Consider a Poisson point process with unknown support boundary curve g, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form ( )integral Phi (g (x)) dx. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Holder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on Phi which are satisfied for Phi (u) = vertical bar u vertical bar(p), p >= 1. As an application, we consider the problem of estimating the L-p-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.