Density estimation with contamination: minimax rates and theory of adaptation

Abstract

This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_{0})$ at some $x_{0}\in\mathbb{R}$ with i.i.d. contaminated observations: \[X_{1},\dots,X_{n}\sim (1-\epsilon)f+\epsilon g\] where $g$ stands for a contamination distribution. We closely track the effect of contamination by the following model index: contamination proportion $\epsilon$, smoothness of the target density $\beta_{0}$, smoothness of the contamination density $\beta_{1}$, and the local level of contamination $m$ such that $g(x_{0})\leq{m}$. The local effect of contamination is shown to depend intricately on the interplay of these parameters. In particular, under a minimax framework, the cost \[[\epsilon^{2}(1\wedge m)^{2}]\vee[n^{-\frac{2\beta_{1}}{2\beta_{1}+1}}\epsilon^{\frac{2}{2\beta_{1}+1}}]\] is shown to be the optimal cost for contamination compared with the usual minimax rate without contamination. The lower bound relies on a novel construction that involves perturbations of a density function at two different resolutions. Such a construction may be of independent interest for the study of local effect of contamination in other nonparametric estimation problems. We also study the setting without any assumption on the contamination distribution, and the minimax cost for contamination is shown to be \[\epsilon^{\frac{2\beta_{0}}{\beta_{0}+1}}.\] Finally, the minimax cost for adaptation is established both for smooth contamination and arbitrary contamination. Under arbitrary contamination, we show that while adaptation to either contamination proportion or smoothness only costs a logarithmic factor, adaptation to both numbers is impossible.