  Missing data is frequently encountered in practice. Propensity score
estimation is a popular tool for handling such missingness. The propensity
score is often developed using the model for the response 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. By partitioning the sample into two groups based on
the response status of the elements, we can apply the density ratio function
estimation method and obtain the inverse propensity scores for nonresponse
adjustment. Density ratio estimation can be obtained by applying the so-called
maximum entropy method, which uses the Kullback-Leibler divergence measure
under calibration constraints. By including the covariates for the outcome
regression models only into the density ratio model, we can achieve efficient
propensity score estimation. We further extend the proposed approach to the
multivariate missing case. Some limited simulation studies are presented to
compare with the existing methods.
