Estimation of sums of random variables: Examples and information bounds

The Annals of Statistics · 2005 · Volume 33 · Issue 5

Abstract

This paper concerns the estimation of sums of functions of observable and unobservable variables. Lower bounds for the asymptotic variance and a convolution theorem are derived in general finite- and infinite-dimensional models. An explicit relationship is established between efficient influence functions for the estimation of sums of variables and the estimation of their means. Certain “plug-in” estimators are proved to be asymptotically efficient in finite-dimensional models, while “u,v” estimators of Robbins are proved to be efficient in infinite-dimensional mixture models. Examples include certain species, network and data confidentiality problems.

Keywords

62F10, 62F12, 62F15, 62G05, 62G20, data confidentiality, Disclosure risk, efficient estimation, Empirical Bayes, influence function, information bound, networks, node degree, species problem, sum of variables, Utility

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