Nonparametric Smoothing Estimation of Conditional Distribution Functions with Longitudinal Data and Time-Varying Parametric Models

The thesis is concerned with the nonparametric estimation of the conditional distribution function with longitudinal data. Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. In this Dissertation, we propose a nonparametric approach based on time-varying parametric models for smoothing estimation of the conditional distribution functions with a longitudinal sample and show that our local polynomial smoothing estimator outperforms the existing Nadaraya-Watson kernel smoothing estimator in term of root MSE and length of condence band. In both cases, we have used the Epanechnikov kernel and bandwidth 2.5.;Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after log transformation or local Box-Cox transformation, but the parameters are smooth function of time. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Pointwise bootstrap condence bands have been constructed for both local polynomial smoothing estimators and Nadaraya-Watson kernel smoothing estimators, resulting in a wider bootstrap condence band for the Nadaraya-Wastson kernel smoothing estimator. Asymptotic properties, including the asymptotic biases, variances and mean squared errors, have been derived for the local polynomial smoothed estimators. Asymptotic distribution of the raw estimators of the conditional distribution functions has been derived.;Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. In our NGHS (National Health and Growth Study) application, we report that (a) Structural Nonparametric Model (SNM) performs better than the Unstructured Nonparametric Model (UNM) in estimating raw probabilities as well as smoothing probabilities on entire time design points. (b) African American (AA) girls have higher probability of developing hypertension than the Caucasian (CC) girls. (c) Box-Cox transformation gives better results than the Log transformation. (d) Smoothing-Ealry and Smoothing-Later give the same results when Log transformation is involved. (e) Smoothing-Later is the only option when Box-Cox transformation is involved.;Finite sample properties of our procedures are investigated through a simulation study. We report that root MSE is smaller at each of the 101 time design point for local polynomial smoothing estimator than the Nadaraya-Watson kernel smoothing estimator. A much stronger conclusion for smaller root MSE is demonstrated by structural nonparametric model than the unstructured nonparametric model when extreme conditional tail probabilities are estimated and smoothed.