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On minimax density estimation on \(\mathbb R\). (English) Zbl 1076.62037

Summary: The problem of density estimation on \(\mathbb R\) on the basis of an independent sample \(X_1, \dots, X_N\) with common density \(f\) is discussed. The behaviour of the minimax \(L_p\) risk, \(1\leq p\leq \infty\), is studied when \(f\) belongs to a Hölder class of regularity \(s\) on the real line. The lower bound for the minimax risk is given. We show that the linear estimator is not efficient in this setting and construct a wavelet adaptive estimator which attains (up to a logarithmic factor in \(N\)) the lower bounds involved. We show that the minimax risk depends on the parameter \(p\) when \(p< 2+1/s\).

MSC:

62G07 Density estimation
62C20 Minimax procedures in statistical decision theory
65T60 Numerical methods for wavelets
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