Density Estimation with Contaminated Data: Minimax Rates and Theory of Adaptation
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. observations, $$ X_1,\dots,X_n\sim (1-ε)f+εg, $$ where $g$ stands for a contamination distribution. In the context of multiple testing, this can be interpreted as estimating the null density at a point. We carefully study the effect of contamination on estimation through the following model indices: contamination proportion $ε$, smoothness of target density $β_0$, smoothness of contamination density $β_1$, and level of contamination $m$ at the point to be estimated, i.e. $g(x_0)\leq m$. It is shown that the minimax rate with respect to the squared error loss is of order $$ [n^{-\frac{2β_0}{2β_0+1}}]\vee[ε^2(1\wedge m)^2]\vee[n^{-\frac{2β_1}{2β_1+1}}ε^{\frac{2}{2β_1+1}}], $$ which characterizes the exact influence of contamination on the difficulty of the problem. We then establish the minimal cost of adaptation to contamination proportion, to smoothness and to both of the numbers. It is shown that some small price needs to be paid for adaptation in any of the three cases. Variations of Lepski's method are considered to achieve optimal adaptation. The problem is also studied when there is no smoothness assumption on the contamination distribution. This setting that allows for an arbitrary contamination distribution is recognized as Huber's $ε$-contamination model. The minimax rate is shown to be $$ [n^{-\frac{2β_0}{2β_0+1}}]\vee [ε^{\frac{2β_0}{β_0+1}}]. $$ The adaptation theory is also different from the smooth contamination case. While adaptation to either contamination proportion or smoothness only costs a logarithmic factor, adaptation to both numbers is proved to be impossible.