Functional estimation and hypothesis testing in nonparametric boundary models

2017/08/09 by Markus Reiß, Martin Wahl, Reiß, Markus +1
Mathematics · #62G05 #62G10 #62G32 #62M30 #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:62G05 #msc:62G10 #msc:62G32 #msc:62M30 #stat.TH

paper · pdf · doi:10.48550/arxiv.1708.02854

21 pages, 1 figure

arxiv created 2019/02/12 · arxiv updated 2019/02/13

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 ∫ Φ(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 Φ which are satisfied for Φ(u)=|u|^p, p≥ 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.

Related

• Efficient estimation of functionals in nonparametric boundary models
• Hypotheses tests in boundary regression models
• Posterior contraction rates for support boundary recovery
• Nonparametric Poisson regression from independent and weakly dependent observations by model selection
• Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery