Asymptotic distribution theory on pseudo semiparametric maximum likelihood estimator with covariates missing not at random

Recently, Cook et al. proposed a semiparametric likelihood estimator to improve study efficiency for a kind of survival data with covariate entries missing not at random (MNAR). Readily available supplementary information on the covariate is utilized in the estimation. They assume that the conditional distributions of the covariate X that having missing entry given the completely observed covariate Z, is known. Guo et al. suggested to replace with its consistent estimator in the likelihood equation when is unknown. However, they did not derive the asymptotic theory of the resulted estimator in this case. This paper fills the gap. The theoretical development makes use of the theory of modern empirical process.

Automated summary

In this study, a semiparametric likelihood estimator was proposed to improve study efficiency for survival data with non-random missing covariate entries. The estimation utilizes supplementary information on the covariate and fills the gap in deriving the asymptotic theory of the resulting estimator. The theoretical development leverages the theory of modern empirical process.