In this dissertation, we consider nonparametric and semiparametric regression for both independent and longitudinal data with missing at random (MAR). The thesis consists of three chapters. In chapter 1, we focus on nonparametric regression of a scalar outcome on a covariate when the outcome is MAR. We show that the usual nonparametric kernel regression estimation based only on complete cases is generally inconsistent. We propose inverse probability weighted (IPW) kernel estimating equations (KEEs) and a class of augmented IPW (AIPW) KEEs. Both approaches do not require specification of a parametric model for the error distribution. We show that the IPW kernel estimator is consistent when the probability that a sampling unit is observed, i.e., the selection probability, is known by design or is estimated using a correctly specified model. We further show that the AIPW kernel estimator is double-robust in the sense that it is consistent if either the model for the selection probability or the model for the conditional mean of the outcome given covariates and auxiliary variables is correctly specified, not necessarily both. We argue that adequate augmentation terms in the AIPW KEEs help increase the efficiency of the estimator. We study the asymptotic properties of the proposed IPW and AIPW kernel estimators, perform simulations to evaluate their finite sample performance, and apply to the analysis of the AIDS Costs and Services Utilization Survey data.

In chapter 2, we consider semiparametric generalized partial linear regression models when the outcome is MAR. We propose a class of AIPW kernel-profile estimating equations, where the nonparametric parameter is estimated using AIPW KEEs and the parametric regression coefficients are estimated using AIPW profile estimating equations. The AIPW kernel-profile estimating equations require input estimates of the selection probabilities and of the conditional mean of the outcome given covariates and auxiliaries under working parametric models. We show that the AIPW estimators of both nonparametric and parametric components are double-robust, i.e. they are consistent provided one of the working models is correct, not necessarily both. In addition, the AIPW estimator of the parametric component is asymptotically normal and locally semiparametric efficient. We conduct simulations to evaluate the finite sample performance, and apply to data to investigate the risk factors of myocardial ischemia. We consider in Chapter 3 nonparametric regression for longitudinal data when some subjects drop out at random. We propose IPW kernel generalized estimating equations (GEEs) and IPW seemingly unrelated (SUR) KEEs using either complete cases or all available cases. Using all available cases help to gain efficiency compared to using complete cases when appropriate covariance matrices are used. We show that these estimators are all consistent when the probabilities of dropout are known or estimated using correctly specified parametric models. The most efficient IPW kernel GEE estimator is obtained by ignoring the within-subject correlation, while the most efficient IPW SUR kernel estimator is obtained by accounting for the within-subject correlation and is more efficient than the most efficient IPW kernel GEE counterpart. We perform simulations to evaluate their finite sample performance.