Efficient estimation of functionals in nonparametric boundary models

TL;DR

This paper develops efficient, unbiased estimators for linear functionals in nonparametric boundary models with one-sided errors, achieving minimax optimal rates and non-asymptotic efficiency, supported by theoretical proofs and simulations.

Contribution

It introduces a simple blockwise estimator and a nonparametric maximum-likelihood approach that attain minimax optimality and UMVU efficiency in boundary models.

Findings

Achieved minimax optimal rates for linear functional estimation.

Constructed UMVU estimators with non-asymptotic efficiency.

Validated estimators through simulations confirming practical applicability.

Abstract

For nonparametric regression with one-sided errors and a boundary curve model for Poisson point processes we consider the problem of efficient estimation for linear functionals. The minimax optimal rate is obtained by an unbiased estimation method which nevertheless depends on a H\"older condition or monotonicity assumption for the underlying regression or boundary 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).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.