Efficient Estimation of Monotone Boundaries

Let g: 0, 1 → 0, 1 be a monotone nondecreasing function and let G be the closure of the set \(x, y) ∈ 0, 1 × 0, 1: 0 ≤ y ≤ 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.