## The Annals of Statistics

### Efficient Estimation of Linear Functionals of a Probability Measure $P$ with Known Marginal Distributions

#### 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."

#### Article information

Source
Ann. Statist., Volume 19, Number 3 (1991), 1316-1346.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348251

Digital Object Identifier
doi:10.1214/aos/1176348251

Mathematical Reviews number (MathSciNet)
MR1126327

Zentralblatt MATH identifier
0742.62034

JSTOR
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. 19 (1991), no. 3, 1316--1346. doi:10.1214/aos/1176348251. https://projecteuclid.org/euclid.aos/1176348251