Estimating nonparametric functionals efficiently under one-sided errors


Séminaire Probabilités & Statistique

22/01/2015 - 14:00 Mr Markus REISS (Humbolt-Universität zu Berlin (Allemagne)) Salle 1 - Tour IRMA

For nonparametric regression with one-sided errors and a related
continuous-time model for Poisson point processes we consider the problem
of efficient estimation for linear functionals of the regression function.
The optimal rate is obtained by an unbiased estimation method which
nevertheless depends on a H"older condition or monotonicity assumption
for the underlying regression function.
We first construct a simple blockwise estimator and then build up a
nonparametric maximum-likelihood approach for exponential noise variables
and the point process model. In that approach also non-asymptotic
efficiency is obtained (UMVU: uniformly minimum variance among all
unbiased estimators). In addition, under monotonicity the estimator is
automatically rate-optimal and adaptive over H"older classes. The proofs
rely essentially on martingale stopping arguments for counting processes
and the point process geometry. The estimators are easily computable and a
small simulation study confirms their applicability.
(joint with L. Selk)