2019/01/03 by Maurilio Gutzeit, Gutzeit, Maurilio
Mathematics · #62G10 #Ball (mathematics) #Combinatorics #FOS: Mathematics #Function (biology) #Mathematical Approximation and Integration #Mathematical analysis #Mathematical optimization #Mathematics #Minimax #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Smoothness #Sobolev space #Statistics Theory (math.ST) #Type (biology) #math.ST #msc:62G10 #stat.TH
paper · pdf · doi:10.48550/arxiv.1901.00880
openalex publication_date 2019/01/03 · openalex created_date 2019/01/11 · arxiv created 2020/02/17 · arxiv updated 2020/02/19 · openalex updated_date 2026/07/28
In this paper we study the problem of testing if an L2-function f belonging to a certain l2-Sobolev-ball Bt(R) of radius R>0 with smoothness level t>0 indeed exhibits a higher smoothness level s>t, that is, belongs to Bs(R). We assume that only a perturbed version of f is available, where the noise is governed by a standard Brownian motion scaled by (1)/(√(n)). More precisely, considering a testing problem of the form H0:~f∈ Bs(R)~~vs.~~H1:~f∈ Bt(R),~infh∈ Bs\Vert f-h\VertL2>ρ for some ρ>0, we approach the task of identifying the smallest value for ρ, denoted ρ^∗, enabling the existence of a test φ with small error probability in a minimax sense. By deriving lower and upper bounds on ρ^∗, we expose its precise dependence on n: ρ^∗∼ n-(t)/(2t+1/2). As a remarkable aspect of this composite-composite testing problem, it turns out that the rate does not depend on s and is equal to the rate in signal-detection, i.e. the case of a simple null hypothesis.