MINIMAX THEORY OF ESTIMATION OF LINEAR FUNCTIONALS OF THE DECONVOLUTION DENSITY WITH OR WITHOUT SPARSITY

• 成果类型: Article
• 署名作者: Pensky, Marianna
• 署名单位: State University System of Florida; University of Central Florida
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1498
• 发表日期: 2017
• 页码: 1516-1541
• 关键词: higher criticism; mixtures

摘要

The present paper considers the problem of estimating a linear functional Phi = integral(infinity)(-infinity) phi(x) f (x) dx of an unknown deconvolution density f on the basis of n i. i. d. observations, Y-1,..., 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) Sigma(n)(i=1) phi(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 phi. Furthermore, using technique of inversion formulas, we extend the theory to a number of situations when the Fourier transform of phi 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 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.