Open Access
November 2019 Functional estimation and hypothesis testing in nonparametric boundary models
Markus Reiß, Martin Wahl
Bernoulli 25(4A): 2597-2619 (November 2019). DOI: 10.3150/18-BEJ1064


Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int\Phi(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $\Phi$ which are satisfied for $\Phi(u)=|u|^{p}$, $p\ge1$. As an application, we consider the problem of estimating the $L^{p}$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.


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Markus Reiß. Martin Wahl. "Functional estimation and hypothesis testing in nonparametric boundary models." Bernoulli 25 (4A) 2597 - 2619, November 2019.


Received: 1 August 2017; Revised: 1 June 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110106
MathSciNet: MR4003559
Digital Object Identifier: 10.3150/18-BEJ1064

Keywords: minimax hypothesis testing , non-linear functionals , Poisson point process , support estimation

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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