Topics in Semiparametric Regression Estimation with Missing Covariates Using Single-Index Models

Abstract

Missing data are very common in many areas such as sociology, biomedical sciences and clinical trials. Simply ignoring the incomplete cases may cause bias in estimation procedures. In this dissertation we investigate semiparametric estimation of linear regression coefficients through generalized estimating equations with single-index models when some covariates are missing at random for both independent and identically distributed (i.i.d.) data and longitudinal data. Existing popular semiparametric estimators by weighted estimating equations may run into difficulties when some selection probabilities are small or the dimension of the covariates is not low. For i.i.d. data, we propose a new simple parameter estimator using a kernel assisted estimator for the augmentation by a single-index model without using the inverse of selection probabilities. We explore the asymptotic efficiency of the proposed estimator and its relationships with existing estimators. In particular, we show that under certain conditions the proposed estimator is as efficient as the existing methods based on standard kernel smoothing, which are often practically infeasible in the case of multiple covariates. For incomplete longitudinal data, we propose a similar estimator when the covariate is nonmonotone missing at random. Heteroscedasticity is considered and working independence correlation structure is applied to simplify the estimation procedure. Asymptotic consistency and normality are derived along with sandwich formulas for asymptotic covariances. The above methods are supported by simulation studies and real data examples. The numerical results show that the proposed estimators avoid some numerical issues caused by estimated small selection probabilities that are needed in other estimators.