Estimation of the support of the density and its boundary using Random Polyhedron
Résumé
We consider random samples in $\mathbb{R}^d$ drawn from an unknown density. When the support is assumed to be convex and with sharp boundary, the convex hull is an estimator of the support that converges to $S$ with a rate of $n^{-2/(d+1)}$. When the boundary of the support is sharp but the support is no longer assumed to be convex, the usual support estimators converges with a rate of $n^{-1/d}$ or $(\ln(n)/n)^{-1/d}$. This paper is devoted to presenting some new estimators of the support of the density, which are based on some local convexity criteria and converge to $S$ with a rate of $(n/\ln n)^{-2/(d+1)}$ (and their boundary converges toward $\partial S$ with the same rate) when the support is assumed to have a sharp $\mathcal{C}^2$ boundary. The convergence rate is also given when the sharpness hypothesis is relaxed (and it is close to the optimal rate when the dimension is two).
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