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September, 1991 Efficient Estimation of Linear Functionals of a Probability Measure $P$ with Known Marginal Distributions
Peter J. Bickel, Ya'Acov Ritov, Jon A. Wellner
Ann. Statist. 19(3): 1316-1346 (September, 1991). DOI: 10.1214/aos/1176348251

Abstract

Suppose that $P$ is the distribution of a pair of random variables $(X, Y)$ on a product space $\mathbb{X} \times \mathbb{Y}$ with known marginal distributions $P_X$ and $P_Y$. We study efficient estimation of functions $\theta(h) = \int h dP$ for fixed $h: \mathbb{X} \times \mathbb{Y} \rightarrow R$ under iid sampling of $(X, Y)$ pairs from $P$ and a regularity condition on $P$. Our proposed estimator is based on partitions of both $\mathbb{X}$ and $\mathbb{Y}$ and the modified minimum chi-square estimates of Deming and Stephan (1940). The asymptotic behavior of our estimator is governed by the projection on a certain sum subspace of $L_2(P)$, or equivalently by a pair of equations which we call the "ACE equations."

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Peter J. Bickel. Ya'Acov Ritov. Jon A. Wellner. "Efficient Estimation of Linear Functionals of a Probability Measure $P$ with Known Marginal Distributions." Ann. Statist. 19 (3) 1316 - 1346, September, 1991. https://doi.org/10.1214/aos/1176348251

Information

Published: September, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0742.62034
MathSciNet: MR1126327
Digital Object Identifier: 10.1214/aos/1176348251

Subjects:
Primary: 62G05
Secondary: 60F05, 60G44, 62G30

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.19 • No. 3 • September, 1991
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