# Goodness-of-Fit Testing for Hölder-Continuous Densities: Sharp Local Minimax Rates

We consider the goodness-of fit testing problem for Hölder smooth densities over ℝ^d: given n iid observations with unknown density p and given a known density p_0, we investigate how large ρ should be to distinguish, with high probability, the case p=p_0 from the composite alternative of all Hölder-smooth densities p such that p-p_0_t ≥ρ where t ∈ [1,2]. The densities are assumed to be defined over ℝ^d and to have Hölder smoothness parameter α>0. In the present work, we solve the case α≤ 1 and handle the case α>1 using an additional technical restriction on the densities. We identify matching upper and lower bounds on the local minimax rates of testing, given explicitly in terms of p_0. We propose novel test statistics which we believe could be of independent interest. We also establish the first definition of an explicit cutoff u_B allowing us to split ℝ^d into a bulk part (defined as the subset of ℝ^d where p_0 takes only values greater than or equal to u_B) and a tail part (defined as the complementary of the bulk), each part involving fundamentally different contributions to the local minimax rates of testing.

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