Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity

The present paper considers a problem of estimating a linear functional $\Phi=\int_{-\infty}^\infty \varphi(x) f(x)dx$ of an unknown deconvolution density $f$ on the basis of i.i.d. observations $Y_i = \theta_i + \xi_i$ where $\xi_i$ has a known pdf $g$ and $f$ is the pdf of $\theta_i$. Although various aspects and particular cases of this problem have been treated by a number of authors, there are still many gaps. In particular, there are no minimax lower bounds for an estimator of $\Phi$ in the case of an arbitrary function $\varphi$. The general upper bounds for the risk cover only the case when Fourier transform of $\varphi$ exists. Moreover, no theory exists for estimating $\Phi$ in the case when vector $\theta = (\theta_1,\cdot,\theta_n)$ is sparse. In addition, until now, the related problem of estimation of functionals $\Phi_n = n^{-1} \sum_{i=1}^n \varphi(\theta_i)$ in indirect observations have been treated as a separate problem with no connection to estimation of $\Phi$. The objective of the present paper is to fill in the gaps and to develop the general minimax theory of estimation of $\Phi$ and to relate this problem to estimation of $\Phi_n$. We offer a general approach to estimation of $\Phi$ (and $\Phi_n$) and provide the upper and the minimax lower bounds for the risk in the case when function $\varphi$ is square integrable. Furthermore, we extend the theory to a number of situations when Fourier transform of $\varphi$ does not exist. Finally, we generalize our results to handle the situation when vector $\theta$ is sparse. As a direct application of the proposed theory, we automatically recover and extend multiple recent results for a variety of problems such as estimation of the mixing cdf, estimation of the mixing pdf with classical and Berkson errors and estimation of the first absolute moment based on indirect observations.