Exact adaptive pointwise estimation on Sobolev classes of densities. (English) Zbl 0990.62032
Summary: The subject of this paper is to estimate adaptively the common probability density of \(n\) independent, identically distributed random variables. The estimation is done at a fixed point \(x_0\in \mathbb{R}\), over density functions that belong to the Sobolev class \(W_n(\beta, L)\). We consider the adaptive problem setup, where the regularity parameter \(\beta\) is unknown and varies in a given set \(B_n\). A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence, is found.
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