Empirical Likelihood Analysis Of Longitudinal Data And Likelihood-based Dantzig Selector

In this thesis, we arc mainly interested in the analysis of longitudinal data by empirical likelihood method with a thorough consideration of the within-subject cor-relations and the variant selection of the high-dimensional data analysis.Longitudinal data sets arc comprised of repeated observations of an outcome vari-able and a set of covariates for each of many subjects. The key feature of longitudinal data is that longitudinal data arc usually correlated within a subject and independent between subjects. A challenge for longitudinal data analysis is how to account for within-subject correlations. The empirical likelihood method provide a means of de-termining nonparametric confidence regions for statistical functionals. The method is likelihood based, but does not require the assumption of a parametric family of data. The empirical likelihood method has many well known advantages over normal approx-imation based method and the bootstrap method for constructing confidence regions. In Chapter 2 we shall consider how to construct confidence regions for regression co-efficients in a partially linear model of longitudinal data. We use the profile empirical likelihood method. Then main contribution is that we take the within-subject corre-lation into account to improve estimation efficiency. The idea is as follows. First, we suppose a semi-parametric structure for the covariances of observation errors in each subject. Then, we employ both the first order and the second order moment conditions of the observations to construct the estimating equations. The nuisance parameters are profiled away. Since there are nonparametric estimations in the estimating equa-tions, we employ under-smoothing technique to ensure that the empirical log-likelihood ratio statistic tends to a standardχp2 variable in distribution. We conduct thorough simulation studies to illustrate the superiority of our method.Driven by a wide range of practical applications, the study in high-dimensional data analysis make great developments over the last few years. Traditional statistical methods designed for low-dimensional problems become inadequate or break down. The main issue of high dimensional data analysis is how to conduct dimension reduc-tion utilizing certain structure of the data. If many of the predictors are redundant, effectively identifying the subset of important predictors can encourage a more inter-pretable and therefor in practice more useful model. In Chapter 3, we consider the variable selection problem in likelihood setting. The Dantzig selector has received a considerable amount of attention since it was proposed. It was designed for linear regression models when the dimension of the parameter p is large but the set of coeffi-cients is sparse. We extend the idea of variable selection behind Dantzig selector (DS) method to the setting of general likelihood. By taking a linear approximation to the score function, we get a linear structure similar to that in the DS method. Hence the existing results for linear models can be utilized. This method works well even when the number of parameters tends to infinity as sample size increases. We study the existence and uniqueness of the solution. Consistency and asymptotic normality of the estimator arc established. To ensure the consistency of model selection, we propose adaptive Dantzig selector (ADS) method and obtain its oracle property. In the end of Chapter 3,a simulation study is reported to assess the performance of the likelihood based Dantzig selector.In practice, grouped variables often appear in high dimensional problems. The most common examples are the multi-factor ANOVA problem and the additive model with polynomial or nonparametric components. In both situations, variable selection amounts to the selection of groups of variables rather than individual variables. Since there's group structure in these variable selection problems, how to utilize this infor-mation is the key factor when dealing such problems. In Chapter 4, we propose an augmented version of group Dantzig selector to take into account the possible different number of elements in each group. We obtain the un-asymptotic l2-norm bound of the estimate. We also conduct intensive simulations to investigate the behavior of the group Dantzig. Our experiments show that when the group structure we know is cor-rect, the behavior of group Dantzig is superior to the Dantzig, while when the group information is not correct, the performance of group Dantzig deteriorates.