# 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.

READ FULL TEXT
Comments

There are no comments yet.