Some contributions to nonparametric modeling with correlated data

In many statistical applications, data are collected over time and are likely correlated. Theory and empirical studies have shown that ignoring the underlying correlation structure may lead to a less accurate local smoothing estimator. In this dissertation, we investigate how to incorporate the correlation information into the estimation of the nonparametric regression model and the varying-coefficient model. Under the assumption that the error process is an auto-regressive (AR) process, we propose profile least squares techniques to estimate the mean function in the nonparametric regression model and the functional coefficients in the varying-coefficient model respectively. The asymptotic distribution of the proposed estimator under regularity conditions shows that the profile least squares method is asymptotically as efficient as the local linear method with i.i.d. data. Further, we apply the SCAD variable selection procedure (Fan and Li, 2001) to select the order of the AR error process. Extensive Monte Carlo simulation studies are conducted to compare the finite sample performance of the proposed procedures with the existing methods. The simulation results show that the newly proposed procedures can dramatically improve the accuracy of the naive local linear estimates with a working-independent error structure. We also apply the proposed methodology to two real data sets from economic and environmental disciplines. In addition, we extend the profile least squares estimation to nonparametric regression with multiple responses. The simulation results imply that our method can work with multiple responses as well.