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Estimation of sums of random variables: examples and information bounds. (English) Zbl 1086.62035
Summary: This paper concerns the estimation of sums of functions of obsersable 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 the “\(u,v\)” estimators of H. Robbins [Statistical Decision Theory and Related Topics 4, 4th Purdue Symp., West Lafayette/Indiana 1986, Vol.1, 265–269 (1988; Zbl 0685.62032)] are proved to be efficient in infinite-dimensional mixture models. Examples include certain species, network and data confidentiality problems.

62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
62P99 Applications of statistics
62F10 Point estimation
62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
62P10 Applications of statistics to biology and medical sciences; meta analysis
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