On minimax density estimation on R
Anatoli Juditsky, Sophie Lambert‐Lacroix · 2001
Abstract: the problem of density estimation on R from an independent sample X1,...XN with common density f is concerned. The behavior of the minimax Lp-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 provided. 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. Key words: nonparametric density estimation, minimax estimation, adaptive estimation. 1