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

## TL;DR

This paper advances adaptive minimax estimation in the convolution structure density model over anisotropic Nikol'skii classes, highlighting the impact of boundedness on risk and proposing an near-optimal adaptive estimator.

## Contribution

It fully characterizes the minimax risk behavior across different parameters and introduces a selection rule for constructing nearly optimal adaptive estimators.

## Key findings

- Boundedness of the function improves minimax risk asymptotics.
- The proposed selection rule yields near-optimal adaptive estimators.
- The behavior of minimax risk varies with regularity and norm parameters.

## Abstract

This paper continues the research started in \cite{LW16}. 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.