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 never-theless depends on a Hölder 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 vari-ables and the point process model. In that approach also non-asymptotic efficiency is obtained (UMVU: uniformly minimum variance among all un-biased estimators). In addition, under monotonicity the estimator is auto-matically rate-optimal and adaptive over Hölder 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. Key words and Phrases: frontier estimation, support estimation, Poisson point process, sufficiency, completeness, UMVU, nonparametric MLE, shape constraint, monoton