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

Abstract

The present paper considers a problem of estimating a linear functional Φ=φ(x)f(x)dx\Phi=\int_{-\infty}^\infty \varphi(x) f(x)dx of an unknown deconvolution density ff on the basis of i.i.d. observations Yi=θi+ξiY_i = \theta_i + \xi_i where ξi\xi_i has a known pdf gg and ff is the pdf of θi\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 Φ\Phi for an arbitrary function φ\varphi. The general upper risk bounds cover only the case when the Fourier transform of φ\varphi exists. Moreover, no theory exists for estimating Φ\Phi when vector of observations is sparse. In addition, until now, the related problem of estimation of functionals Φn=n1i=1nφ(θi)\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 Φ\Phi. The objective of the present paper is to fill in the gaps and develop the general minimax theory of estimation of Φ\Phi and Φn\Phi_n. We offer a general approach to estimation of Φ\Phi (and Φn\Phi_n) and provide the upper and the minimax lower risk bounds in the case when function φ\varphi is square integrable. Furthermore, we extend the theory to the case when Fourier transform of φ\varphi does not exist and Φ\Phi can be presented as a linear functional of the Fourier transform of ff and its derivatives. Finally, we generalize our results to handle the situation when vector θ\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)(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.

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