Minimax L_2-Separation Rate in Testing the Sobolev-Type Regularity of a function
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In this paper we study the problem of testing if an L_2-function f belonging to a certain l_2-Sobolev-ball B_t of radius R>0 with smoothness level t>0 indeed exhibits a higher smoothness level s>t, that is, belongs to B_s. 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 H_0: f∈ B_s vs. H_1: _h∈ B_s f-h_L_2>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.
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