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) is an element of [0,1] X [0,1]: 0 less than or equal to y less than or equal to 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.