Efficient Estimation of Monotone Boundaries
A. P. Korostelev, L. Simar, A. B. Tsybakov
The Annals of Statistics · 1995 · Volume 23 · Issue 2
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.
Keywords
62G05; 62G20; efficiency; estimation of support of a density; free disposal hull; Hausdorff distance; minimum risk; Monotone boundary;
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