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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.

In observational studies subjects may self select, thereby creating a biased sample. Such problems arise frequently, for example, in astronomical, biomedical, animal, and oil studies, survey sampling and econometrics. For a typical subject, let Y denote the value of interest and suppose that Y has an unknown density function f . Further, let w(y) denote the probability that the subject includes itself in the study given Y = y. Then the conditional density of Y given that it is observed is f (y) = w(y)f(y)=, where is a normalizing constant. The problem of estimating w and f from a biased sample X 1 ; : : : ; X n independently from f is considered when f is known to belong to a parametric family, say f = f ` , where ` is a vector of unknown parameters, and w is assumed to be non-decreasing. An algorithm for computing the maximum likelihood estimator of (w; `) is developed, and consistency is established. Simulations are used to show that our method is feasible with moderate sam...

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This paper evaluates how five state pre-kindergarten (pre-K) programs affected children’s receptive vocabulary, math, and print awareness skills. Taking advantage of each state’s strict enrollment policy determined by a child’s date of birth, a regression-discontinuity design was

Estimating the mean of high valued observations