Optimal pseudolikelihood estimation in the analysis of multivariate missing data with nonignorable nonresponse

Tang et al. (2003) considered a regression model with missing response, where the missingness mechanism depends on the value of the response variable and hence is nonignorable. They proposed three pseudolikelihood estimators, based on different treatments of the probability distribution of the completely observed covariates. The first assumes the distribution of the covariate to be known, the second estimates this distribution parametrically, and the third estimates the distribution nonparametrically. While it is not hard to show that the second estimator is more efficient than the first, Tang et al. (2003) only conjectured that the third estimator is more efficient than the first two. In this paper, we investigate the asymptotic behaviour of the third estimator by deriving a closed-form representation of its asymptotic variance. We then prove that the third estimator is more efficient than the other two. Our result can be straightforwardly applied to missingness mechanisms that are more general than that in Tang et al. (2003).