Efficient Estimation of Monotone Boundaries

Korostelev, AP.;Simar, Léopold;Tsybakov, AB.
(1995) Annals of Statistics — Vol. 23, n° 2, p. 476-489 (1995)

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Abstract
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.
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Korostelev, AP., Simar, L., & Tsybakov, AB. (1995). Efficient Estimation of Monotone Boundaries. Annals of Statistics, 23(2), 476-489. https://doi.org/10.1214/aos/1176324531 (Original work published 1995)