Functional estimation and hypothesis testing in nonparametric boundary models

TL;DR

This paper develops a nonparametric estimator for boundary models based on Poisson processes, achieving optimal convergence rates and applying it to $L^p$-norm estimation and hypothesis testing.

Contribution

It introduces a UMVU estimator for nonlinear functionals in boundary models that attains minimax rates under weak conditions, extending nonparametric inference methods.

Findings

Estimator is UMVU over H"older balls.

Achieves local minimax convergence rate.

Derives minimax separation rates for hypothesis testing.

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 H\"older balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $\Phi$ which are satisfied for $\Phi(u)=|u|^p$, $p\ge 1$. 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.