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.

翻译：本文采用简易经验似然方法，在存在辅助信息的条件下构建了概率测度线性泛函的半参数有效估计量。我们使用估计约束函数，并允许约束数量随样本量增长。研究了三类可由无穷多约束表征的信息情形：（1）边际分布已知；（2）边际分布未知但相同；（3）分布对称性。定义了改进的空间深度函数并研究了其渐近性质。报告了有关效率提升的模拟结果。