Estimation in the convolution structure density model. Part II: adaptation over the scale of anisotropic classes

2017/04/14 by Oleg Lepski, Lepski, Oleg, Thomas Willer +1
Engineering · #FOS: Mathematics #Hydrocarbon exploration and reservoir analysis #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1704.04420

openalex publication_date 2017/04/14 · openalex created_date 2017/04/28 · openalex updated_date 2026/07/28

Abstract

This paper continues the research started in \citeLW16. In the framework of the convolution structure density model on \bR^d, we address the problem of adaptive minimax estimation with \bL_p--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.

Citations

• Some new ideas in nonparametric estimation
• Model selection for density estimation with L2-loss

Related

• On adaptive minimax density estimation on R^d
• Estimation in the convolution structure density model. Part I: oracle inequalities
• Adaptive estimation over anisotropic functional classes via oracle approach
• Structural adaptive deconvolution under L_p-losses
• Adaptive estimation of \mathbbL₂-norm of a probability density and related topics I. Lower bounds