  Causal inference problems have remained an important research topic over the
past several decades due to their general applicability in assessing a
treatment effect in many different real-world settings. In this paper, we
propose two inferential procedures on the average treatment effect (ATE)
through a two-sample pseudo-empirical likelihood (PEL) approach. The first
procedure uses the estimated propensity scores for the formulation of the PEL
function, and the resulting maximum PEL estimator of the ATE is equivalent to
the inverse probability weighted estimator discussed in the literature. Our
focus in this scenario is on the PEL ratio statistic and establishing its
theoretical properties. The second procedure incorporates outcome regression
models for PEL inference through model-calibration constraints, and the
resulting maximum PEL estimator of the ATE is doubly robust. Our main
theoretical result in this case is the establishment of the asymptotic
distribution of the PEL ratio statistic. We also propose a bootstrap method for
constructing PEL ratio confidence intervals for the ATE to bypass the scaling
constant which is involved in the asymptotic distribution of the PEL ratio
statistic but is very difficult to calculate. Finite sample performances of our
proposed methods with comparisons to existing ones are investigated through
simulation studies.
