Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity (1411.1660v3)

Abstract: 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.