## Bernoulli

• Bernoulli
• Volume 23, Number 2 (2017), 1022-1055.

### Efficient estimation of functionals in nonparametric boundary models

#### 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 Hölder 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.

#### Article information

Source
Bernoulli, Volume 23, Number 2 (2017), 1022-1055.

Dates
Revised: June 2015
First available in Project Euclid: 4 February 2017

https://projecteuclid.org/euclid.bj/1486177391

Digital Object Identifier
doi:10.3150/15-BEJ768

Mathematical Reviews number (MathSciNet)
MR3606758

Zentralblatt MATH identifier
1380.62177

#### Citation

Reiß, Markus; Selk, Leonie. Efficient estimation of functionals in nonparametric boundary models. Bernoulli 23 (2017), no. 2, 1022--1055. doi:10.3150/15-BEJ768. https://projecteuclid.org/euclid.bj/1486177391

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