Estimation of linear and nonlinear functionals in nonparametric boundary models

Gwennaëlle Mabon und Markus Reiß (HU Berlin)
Wednesday, November 30, 2016 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

For nonparametric regression with one-sided errors and a boundary curve model for Poisson point R processes we consider first the problem of efficient estimation for linear functionals of the form \int f(x)w(x)dx with unknown f and known w. We propose a simple blockwise estimator and then build up a nonparametric maximum-likelihood approach. Both methods allow for estimation with optimal rate n^{-(\beta+1/2)/(\beta+1) under \beta-Hölder smoothness or monotonicity constraints (analogue of \beta = 1). Surprisingly, the nonparametric MLE approach enjoys additionally non-asymptotic efficiency properties (UMVU: uniformly minimum variance among all unbiased estimators). Given uniform observations supported on a convex set, the theory extends to unbiased estimation of the volume of this convex set.
In a second step, we consider estimation of nonlinear functionals of the form \int \Phi(f(x))dx for known weakly differentiable \Phi. In view of L^p-norms a primary example is \Phi(x) = |x|^p. Even in that case unbiased estimation is feasible and optimal convergence rates can be derived. As an application we discuss L^p-separation rates in nonparametric testing.
The proofs rely essentially on martingale stopping arguments for counting processes and the underlying point process geometry. The estimators are easily computable and some simulation results are presented. Surprising differences with standard models like Gaussian white noise are discussed. Diffculties with establishing a Bayesian Bernstein-von Mises result are pointed out.
(based on joint work with Leonie Selk, Nikolay Baldin and Martin Wahl)