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 $p_0$, we investigate how large $\rho$ should be to distinguish, with high probability, the case $p=p_0$ from the composite alternative 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 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 $p_0$. 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 $\mathbb{R}^d$ into a bulk part (defined as the subset of $\mathbb{R}^d$ 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.

翻译：我们考虑$\mathbb{R}^d$上Hölder光滑密度的拟合优度检验问题：给定来自未知密度$p$的$n$个独立同分布观测，以及已知密度$p_0$，研究$\rho$需多大才能以高概率区分$p=p_0$的情形与所有满足$\|p-p_0\|_t \geq \rho$（其中$t \in [1,2]$）的Hölder光滑密度$p$构成的复合备择假设。假定这些密度定义于$\mathbb{R}^d$上，且具有Hölder光滑性参数$\alpha>0$。在本文中，我们解决了$\alpha \leq 1$的情形，并通过对密度施加额外的技术限制处理了$\alpha>1$的情形。我们给出了检验的局部极小极大速率的匹配上界和下界，这些界以$p_0$显式表示。我们提出了新颖的检验统计量，相信这些统计量可能具有独立的研究价值。我们还首次定义了显式截断值$u_B$，使得能将$\mathbb{R}^d$划分为主体部分（定义为$\mathbb{R}^d$中$p_0$取值大于等于$u_B$的子集）和尾部部分（定义为主体的补集），这两个部分对局部极小极大检验速率的贡献本质不同。