Robust and Efficient Methods in Semi-supervised Inference and Causal Inference

We consider the mean response estimation and inference in semi-supervised settings in the first two chapters. Such settings consist of a relatively small labeled dataset and an extensive unlabeled dataset. Chapter 1 considers the classical semi-supervised setup that the outcome is missing completely at random (MCAR). Our goal is to improve the efficiency of the supervised sample mean estimator using the additional unlabeled data. We proposed a semi-supervised mean estimator based on flexible working models, including high-dimensional and non-parametric models. In Chapter 2, we further consider the situation that a selection bias may appear. Our goal is to remove the bias originating from the dependence between the missing and outcome. We propose a semi-supervised doubly robust mean estimator with valid inference results when some product rate condition holds. Our work fills in the gap between the semi-supervised literature and the missing data literature. We allow selection bias -- this extends the semi-supervised literature. We also allow extremely unbalanced labeled/unlabeled groups and violate the usual positivity condition, which is always assumed throughout the missing data literature.The last two chapters consider the estimation and inference of the dynamic treatment effect (DTE) when the treatment variable is longitudinal and the covariates are possibly high dimensional. Chapter 3 proposes a doubly robust DTE estimator based on (imputed) Lasso-type nuisance estimators. We established root-n inference when all the nuisance models are correctly specified and some sparsity conditions hold. Chapter 4 further provides root-n inference for the DTE even when model misspecification occurs. This is achieved based on special ``moment targeting'' nuisance estimators. We provide valid inference as long as one of the nuisance models is correctly specified at each time spot -- such a result is better than all the existing literature, even containing the low-dimensional works