Bernoulli

  • 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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Abstract

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1568362037

Digital Object Identifier
doi:10.3150/18-BEJ1064

Mathematical Reviews number (MathSciNet)
MR4003559

Zentralblatt MATH identifier
07110106

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

Citation

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. https://projecteuclid.org/euclid.bj/1568362037


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