Title: Density Estimation with Contaminated Data: 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. observations, $$ X_1,\dots,X_n\sim
(1-\epsilon)f+\epsilon g, $$ 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
$\epsilon$, smoothness of target density $\beta_0$, smoothness of contamination
density $\beta_1$, 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 $$
[n^{-\frac{2\beta_0}{2\beta_0+1}}]\vee[\epsilon^2(1\wedge
m)^2]\vee[n^{-\frac{2\beta_1}{2\beta_1+1}}\epsilon^{\frac{2}{2\beta_1+1}}], $$
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 $\epsilon$-contamination
model. The minimax rate is shown to be $$
[n^{-\frac{2\beta_0}{2\beta_0+1}}]\vee [\epsilon^{\frac{2\beta_0}{\beta_0+1}}].
$$ 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.