ESTIMATION IN THE CONVOLUTION STRUCTURE DENSITY MODEL. PART II: ADAPTATION OVER THE SCALE OF ANISOTROPIC CLASSES.

O V Lepski (1) , T Willer (1)

1 I2M - Institut de Mathématiques de Marseille

This paper continues the research started in Lepski and Willer (2016). In the framework of the convolution structure density model on R d , we address the problem of adaptive minimax estimation with Lp–loss over the scale of anisotropic Nikol'skii classes. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. In particular, we show that the boundedness of the function to be estimated leads to an essential improvement of the asymptotic of the minimax risk. We prove that the selection rule proposed in Part I leads to the construction of an optimally or nearly optimally (up to logarithmic factor) adaptive estimator.

Mots clés: deconvolution model; density estimation; oracle inequality; adaptive estimation; kernel estimators; Lp–risk; anisotropic Nikol’skii class

Domaines: Statistiques [math.ST]

O V Lepski, T Willer. ESTIMATION IN THE CONVOLUTION STRUCTURE DENSITY MODEL. PART II: ADAPTATION OVER THE SCALE OF ANISOTROPIC CLASSES.. 2017. ⟨hal-01508694⟩