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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 $\Phi$ for an arbitrary function $\varphi$. The general upper risk bounds cover only the case when the Fourier transform of $\varphi$ exists. Moreover, no theory exists for estimating $\Phi$ when vector of observations 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 develop the general minimax theory of estimation of $\Phi$ and $\Phi_n$. We offer a general approach to estimation of $\Phi$ (and $\Phi_n$) and provide the upper and the minimax lower risk bounds in the case when function $\varphi$ is square integrable. Furthermore, we extend the theory to the case when Fourier transform of $\varphi$ does not exist and $\Phi$ can be presented as a linear functional of the Fourier transform of $f$ and its derivatives. Finally, we generalize our results to handle the situation when vector $\theta$ is sparse. As a direct application of the proposed theory, we obtain multiple new results and automatically recover existing ones for a variety of problems such as estimation of the $(2M+1)$-th absolute moment or a generalized moment of the deconvolution density, estimation of the mixing cdf or estimation of the mixing pdf with classical and Berkson errors.

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