期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2021
卷号：83
期号：2
页码：569-596
DOI：10.1007/s13171-020-00232-1
语种：English
出版社：Indian Statistical Institute
摘要：LetEbe a separable Banach space and letf:E↦ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:E\mapsto {\mathbb {R}}$\end{document}be a smooth functional. We discuss a problem of estimation off(??) based on an observationX=??+ξ, where??∈Eis an unknown parameter andξis a mean zero random noise, or based onni.i.d. observations from the same random shift model. We develop estimators off(??) with sharp mean squared error rates depending on the degree of smoothness offfor random shift models with distribution of the noiseξsatisfying Poincaré type inequalities (in particular, for some log-concave distributions). We show that for sufficiently smooth functionalsfthese estimators are asymptotically normal with a parametric convergence rate. This is done both in the case of known distribution of the noise and in the case when the distribution of the noise is Gaussian with covariance being an unknown nuisance parameter.
