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

We consider the goodness-of fit testing problem for H\"older smooth densities over $\mathbb{R}^d$: given $n$ iid observations with unknown density $p$ and given a known density $p0$, we investigate how large $\rho$ should be to distinguish, with high probability, the case $p=p0$ from the composite alternative of all H\"older-smooth densities $p$ such that $|p-p0|t \geq \rho$ where $t \in [1,2]$. The densities are assumed to be defined over $\mathbb{R}^d$ and to have H\"older smoothness parameter $\alpha>0$. In the present work, we solve the case $\alpha \leq 1$ and handle the case $\alpha>1$ 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 $p0$. We propose novel test statistics which we believe could be of independent interest. We also establish the first definition of an explicit cutoff $uB$ allowing us to split $\mathbb{R}^d$ into a bulk part (defined as the subset of $\mathbb{R}^d$ where $p0$ takes only values greater than or equal to $uB$) and a tail part (defined as the complementary of the bulk), each part involving fundamentally different contributions to the local minimax rates of testing.