Efficient Estimation of Monotone Boundaries

Abstract

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.