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

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