Efficient Estimation of Monotone Boundaries

Let $g: \lbrack 0, 1\rbrack \rightarrow \lbrack 0, 1\rbrack$ be a monotone nondecreasing function and let $G$ be the closure of the set $\{(x, y) \in \lbrack 0, 1\rbrack \times \lbrack 0, 1\rbrack: 0 \leq y \leq g (x)\}$. We consider the problem of estimating the set $G$ from a sample of i.i.d. observations uniformly distributed in $G$. The estimation error is measured in the Hausdorff metric. We propose the estimator which is asymptotically efficient in the minimax sense.