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Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures

08/25/2020
by   Alexander Goldenshluger, et al.
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In this paper we develop rate–optimal estimation procedures in the problem of estimating the L_p–norm, p∈ (0, ∞) of a probability density from independent observations. The density is assumed to be defined on R^d, d≥ 1 and to belong to a ball in the anisotropic Nikolskii space. We adopt the minimax approach and construct rate–optimal estimators in the case of integer p≥ 2. We demonstrate that, depending on parameters of Nikolskii's class and the norm index p, the risk asymptotics ranges from inconsistency to √(n)–estimation. The results in this paper complement the minimax lower bounds derived in the companion paper <cit.>.

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