Estimating nonparametric functionals efficiently under one-sided errors

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.