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

Understand

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 $p_0$.