Efficient Estimation of Monotone Boundaries

Abstract

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.