  We propose a new modeling and estimation approach to select the optimal
treatment regime from different options through constructing a robust
estimating equation. The method is protected against misspecification of the
propensity score model, the outcome regression model for the non-treated group,
or the potential non-monotonic treatment difference model. Our method also
allows residual errors to depend on covariates. A single index structure is
incorporated to facilitate the nonparametric estimation of the treatment
difference. We then identify the optimal treatment through maximizing the value
function. Theoretical properties of the treatment assignment strategy are
established. We illustrate the performance and effectiveness of our proposed
estimators through extensive simulation studies and a real dataset on the
effect of maternal smoking on baby birth weight.
