On minimax density estimation on \mathbb{R}}

Abstract

The problem of density estimation on \mathbb{R}} on the basis of an independent sample X₁,..., X_N with common density f is discussed. The behaviour of the minimax L_p risk, 1≤p≤∞, 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.