Nuisance parameter elimination for proportional likelihood ratio models with nonignorable missingness and random truncation.

We show that the proportional likelihood ratio model proposed recently by Luo & Tsai (2012) enjoys model-invariant properties under certain forms of nonignorable missing mechanisms and randomly double-truncated data, so that target parameters in the population can be estimated consistently from those biased samples. We also construct an alternative estimator for the target parameters by maximizing a pseudo-likelihood that eliminates a functional nuisance parameter in the model. The corresponding estimating equation has a U-statistic structure. As an added advantage of the proposed method, a simple score-type test is developed to test a null hypothesis on the regression coefficients. Simulations show that the proposed estimator has a small-sample efficiency similar to that of the nonparametric likelihood estimator and performs well for certain nonignorable missing data problems.