Measurement error frequently occurs in scientific studies when precise measurements of variables are unavailable or too expensive. It is well-known that in regression models ignoring measurement error leads to biased estimation of regression coefficients. In this thesis, we propose semiparametric methods to correct the bias and improve the efficiency of estimation under two frameworks. Firstly, for proportional hazards models with measurement error in covariates, we propose a new class of semiparametric estimators by solving estimating equations based on the semiparametric efficient scores. The baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are all treated as unknown infinite dimensional. The proposed estimators prove to be locally efficient. Secondly, for a general regression model when error-prone surrogates of true predictors are collected in the primary data set while accurate measurements of the predictors are available only in a small validation data set, we propose a new class of semiparametric estimators for the regression coefficients based on expected estimating equations. The measurement error model is calibrated nonparametrically using a kernel smoothing method. We prove that the proposed estimators are consistent, asymptotically unbiased and normal in both scenarios.

Functional data appear more and more often in scientific fields due to technological advances. In functional data analysis (FDA), function principal components analysis (FPCA) has become one of the most important modeling and dimension reduction tools. Motivated by a recent root image study in plant science where the data have a natural three-level nested hierarchical structure, we analyze the data using multilevel FPCA. We estimate the covariance function of the functional random effects by a fast penalized tensor product spline approach, perform multilevel FPCA using the best linear unbiased predictor of the principal component scores, and improve the estimation efficiency by an iterative algorithm. We choose the number of principal components using a conditional Akaike Information Criterion and test the effect in the mean function using a generalized likelihood ratio test statistic based on the marginal likelihood and the conditional likelihood. Extensive simulation studies have been carried out to evaluate the validity of our proposed methods.