# Estimation in the convolution structure density model. Part I: oracle inequalities

## TL;DR

This paper develops a new pointwise selection rule for kernel estimators in the convolution structure density model, establishing oracle inequalities and demonstrating near-optimal adaptive minimax estimation under $L_p$-loss.

## Contribution

It introduces a novel pointwise selection rule for kernel estimators in the convolution structure density model, with proven oracle inequalities and adaptive minimax optimality results.

## Key findings

- Established $L_p$-norm oracle inequalities for the selected estimator.
- Proved the proposed selection rule yields nearly optimal adaptive estimators.
- Fully characterized the minimax risk behavior over anisotropic Nikol'skii classes.

## Abstract

We study the problem of nonparametric estimation under $\bL_p$-loss, $p\in [1,\infty)$, in the framework of the convolution structure density model on $\bR^d$. This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. In Part I the original pointwise selection rule from a family of "kernel-type" estimators is proposed. For the selected estimator, we prove an $\bL_p$-norm oracle inequality and several of its consequences. In Part II the problem of adaptive minimax estimation under $\bL_p$--loss over the scale of anisotropic Nikol'skii classes is addressed. 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. 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.