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We consider the goodness-of fit testing problem for H\"older smooth densitiesover $\mathbb{R}^d$: given $n$ iid observations with unknown density $p$ andgiven a known density $p_0$, we investigate how large $\rho$ should be todistinguish, with high probability, the case $p=p_0$ from the compositealternative of all H\"older-smooth densities $p$ such that $\|p-p_0\|_t \geq\rho$ 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 thepresent work, we solve the case $\alpha \leq 1$ and handle the case $\alpha>1$using an additional technical restriction on the densities. We identifymatching upper and lower bounds on the local minimax rates of testing, givenexplicitly in terms of $p_0$. We propose novel test statistics which we believecould be of independent interest. We also establish the first definition of anexplicit cutoff $u_B$ allowing us to split $\mathbb{R}^d$ into a bulk part(defined as the subset of $\mathbb{R}^d$ where $p_0$ takes only values greaterthan or equal to $u_B$) and a tail part (defined as the complementary of thebulk), each part involving fundamentally different contributions to the localminimax rates of testing.

Goodness-of-Fit Testing for H\"older-Continuous Densities: Sharp Local Minimax Rates