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."
Publié le : 1991-09-14
Classification:
Marginal distributions,
modified minimum chi square,
alternating projections,
asymptotic normality,
efficiency,
62G05,
60F05,
62G30,
60G44
@article{1176348251,
author = {Bickel, Peter J. and Ritov, Ya'Acov and Wellner, Jon A.},
title = {Efficient Estimation of Linear Functionals of a Probability Measure $P$ with Known Marginal Distributions},
journal = {Ann. Statist.},
volume = {19},
number = {1},
year = {1991},
pages = { 1316-1346},
language = {en},
url = {http://dml.mathdoc.fr/item/1176348251}
}
Bickel, Peter J.; Ritov, Ya'Acov; Wellner, Jon A. Efficient Estimation of Linear Functionals of a Probability Measure $P$ with Known Marginal Distributions. Ann. Statist., Tome 19 (1991) no. 1, pp. 1316-1346. http://gdmltest.u-ga.fr/item/1176348251/