Confidence Sets in Boundary and Set Estimation

Let $p_1\leq p_2$ and consider estimating a fixed set $\{x: p_1 \leq f(x) \leq p_2\}$ by the random set $\{x: p_1\leq \widehat f_n(x)\leq p_2\}$, where $\widehat f_n$ is a consistent estimator of the continuous function $f$. This paper gives consistency conditions for these sets, and provides a new method to construct confidence regions from empirical averages of sets. The method can also be used to construct confidence regions for sets of the form $\{x: f(x)\leq p\}$ and $\{x: f(x)=p\}$. We then apply this approach to set and boundary estimation. We describe conditions for strong consistency for the empirical average sets and study the fluctuations of these via confidence regions. We illustrate the proposed methods on several examples.