2014/11/06 by Marianna Pensky, Pensky, Marianna
Engineering · Mathematics · #Mathematical Approximation and Integration #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #math.ST #msc:62G05 #msc:62G07 #msc:62G20 #stat.TH
paper · pdf · doi:10.48550/arxiv.1411.1660
40 pages
arxiv created 2015/05/15 · arxiv updated 2015/05/19
The present paper considers a problem of estimating a linear functional Φ=∫-∞^∞ φ(x) f(x)dx of an unknown deconvolution density f on the basis of i.i.d. observations Yi = θi + ξi where ξi has a known pdf g and f is the pdf of θi. 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 for an arbitrary function φ. The general upper risk bounds cover only the case when the Fourier transform of φ exists. Moreover, no theory exists for estimating Φ when vector of observations is sparse. In addition, until now, the related problem of estimation of functionals Φn = n-1 ∑i=1n φ(θi) in indirect observations have been treated as a separate problem with no connection to estimation of Φ. The objective of the present paper is to fill in the gaps and develop the general minimax theory of estimation of Φ and Φn. We offer a general approach to estimation of Φ (and Φn) and provide the upper and the minimax lower risk bounds in the case when function φ is square integrable. Furthermore, we extend the theory to the case when Fourier transform of φ does not exist and Φ 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 θ 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.