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

Abstract

The present paper considers the 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 $n$ i.i.d. observations, $Y_{1},\ldots,Y_{n}$ of $Y=\theta+\xi$, where $\xi$ has a known pdf $g$, and $f$ is the pdf of $\theta$. The objective of the present paper is to develop the a general minimax theory of estimating $\Phi$, and to relate this problem to estimation of functionals $\Phi_{n}=n^{-1}\sum_{i=1}^{n}\varphi(\theta_{i})$ in indirect observations. In particular, we offer a general, Fourier transform based approach to estimation of $\Phi$ (and $\Phi_{n}$) and derive upper and minimax lower bounds for the risk for an arbitrary square integrable function $\varphi$. Furthermore, using technique of inversion formulas, we extend the theory to a number of situations when the Fourier transform of $\varphi$ does not exist, but $\Phi$ can be presented as a functional of the Fourier transform of $f$ and its derivatives. The latter enables us to construct minimax estimators of the functionals that have never been handled before such as the odd absolute moments and the generalized moments of the deconvolution density. Finally, we generalize our results to the situation when the vector $\mathbf{{\theta}}$ is sparse and the objective is estimating $\Phi$ (or $\Phi_{n}$) over the nonzero components only. As a direct application of the proposed theory, we automatically recover multiple recent results and obtain a variety of new ones such as, for example, estimation of the mixing probability density function with classical and Berkson errors and estimation of the $(2M+1)$-th absolute moment of the deconvolution density.