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