On adaptive minimax density estimation on R^d

Alexander Goldenshluger (1) , Oleg Lepski (2)

1 Department of Statistics
2 I2M - Institut de Mathématiques de Marseille

We address the problem of adaptive minimax density estimation on $\bR^d$ with $\bL_p$--loss on the anisotropic Nikol'skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in order over the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.

Mots clés: density estimation; oracle inequality; adaptive estimation; kernel estimators; L_p--risk

Domaines: Mathématiques [math]

Alexander Goldenshluger, Oleg Lepski. On adaptive minimax density estimation on R^d. Probability Theory and Related Fields, 2014, 159 (3), pp.479-543. ⟨10.1007/s00440-013-0512-1⟩. ⟨hal-01265245⟩