Robust density estimation over star-shaped density classes

Abstract

We establish a novel criterion for comparing the performance of two densities, $g_1$ and $g_2$, within the context of corrupted data. Utilizing this criterion, we propose an algorithm to construct a density estimator within a star-shaped density class, $\mathcal{F}$, under conditions of data corruption. We proceed to derive the minimax upper and lower bounds for density estimation across this star-shaped density class, characterized by densities that are uniformly bounded above and below (in the sup norm), in the presence of adversarially corrupted data. Specifically, we assume that a fraction $\hat{I}\mu \le \frac{1}{3}$ of the $N$ observations are arbitrarily corrupted. We obtain the minimax upper bound $\max \{\ddot{I}{„}_{\overline{J}}^2,\hat{I}\mu \}\wedge d^2$. Under certain conditions, we obtain the minimax risk, up to proportionality constants, under the squared $L_2$ loss as $$\max \left\{\ddot{I}{„}^{\ast 2}\wedge d^2,\hat{I}\mu \wedge d^2\right\},$$ where $\ddot{I}{„}^{\ast }:=\sup \left\{\ddot{I}„:N\ddot{I}{„}^2\le \log {\mathcal{M}}_{\mathcal{F}}^{\text{loc}}(\ddot{I}„,c)\right\}$ for a sufficiently large constant $c$. Here, ${\mathcal{M}}_{\mathcal{F}}^{\text{loc}}(\ddot{I}„,c)$ denotes the local entropy of the set $\mathcal{F}$, and $d$ is the $L_2$ diameter of $\mathcal{F}$.