• Bernoulli
  • Volume 25, Number 4A (2019), 2597-2619.

Functional estimation and hypothesis testing in nonparametric boundary models

Markus Reiß and Martin Wahl

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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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Bernoulli, Volume 25, Number 4A (2019), 2597-2619.

Received: August 2017
Revised: June 2018
First available in Project Euclid: 13 September 2019

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


Reiß, Markus; Wahl, Martin. Functional estimation and hypothesis testing in nonparametric boundary models. Bernoulli 25 (2019), no. 4A, 2597--2619. doi:10.3150/18-BEJ1064.

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