  Propensity score weighting is widely used to improve the representativeness
and correct the selection bias in the voluntary sample. The propensity score is
often developed using a model for the sampling probability, which can be
subject to model misspecification. In this paper, we consider an alternative
approach of estimating the inverse of the propensity scores using the density
ratio function satisfying the self-efficiency condition. The smoothed density
ratio function is obtained by the solution to the information projection onto
the space satisfying the moment conditions on the balancing scores.
  By including the covariates for the outcome regression models only in the
density ratio model, we can achieve efficient propensity score estimation.
Penalized regression is used to identify important covariates. We further
extend the proposed approach to the multivariate missing case. Some limited
simulation studies are presented to compare with the existing methods.
