  We provide a semi-parametric analysis for the proportional likelihood ratio
model, proposed by Luo & Tsai (2012). We study the tangent spaces for both the
parameter of interest and the nuisance parameter, and obtain an explicit
expression for the efficient score function. We propose a family of
Z-estimators based on the score functions, including an approximated efficient
estimator. Using inverse probability weighting, the proposed estimators can
also be applied to different missing-data mechanisms, such as right censored
data and non-random sampling. A simulation study that illustrates the
finite-sample performance of the estimators is presented.
