# Easy Maximum Empirical Likelihood Estimation of Linear Functionals Of A Probability Measure With Infinitely Many Constraints

## TL;DR

This paper introduces a simple empirical likelihood method for efficiently estimating linear functionals of a probability measure with infinitely many constraints, applicable in various informational settings.

## Contribution

It develops an easy empirical likelihood estimator that handles infinitely many constraints and different types of side information, improving estimation efficiency.

## Key findings

- The estimator achieves semiparametric efficiency.
- Simulation results show significant efficiency gains.
- The method applies to known marginals, unknown identical marginals, and symmetric distributions.

## Abstract

In this article, we construct semiparametrically efficient estimators of linear functionals of a probability measure in the presence of side information using an easy empirical likelihood approach. We use estimated constraint functions and allow the number of constraints to grow with the sample size. Considered are three cases of information which can be characterized by infinitely many constraints: (1) the marginal distributions are known, (2) the marginals are unknown but identical, and (3) distributional symmetry. An improved spatial depth function is defined and its asymptotic properties are studied. Simulation results on efficiency gain are reported.