This dissertation consists of two parts: inferences on semiparametric varying coefficient partially linear models and semilinear high-dimensional models for normalization of microarray data.;Varying-coefficient partially linear models are frequently used in statistical modeling, yet their estimation and inferences have not been systematically studied. Part I proposes a profile least-squares technique for estimating parametric components. The asymptotic normality of the profile least-squares estimator is studied. The main focus is the examination of whether the generalized likelihood techniques that Fan, Zhang and Zhang (2001) develop are applicable to the testing problems in the parametric components of semiparametric models. Profile likelihood ratio tests are introduced. We demonstrate that the profile likelihood ratio statistics are asymptotically distribution-free and follow chi 2-distributions under null hypotheses. This not only unveils a new Wilks type of phenomenon but also provides a simple and useful method for semiparametric inferences. In addition, Wald types of statistics in the semiparametric models are introduced and demonstrated to possess a similar sampling property to profile likelihood ratio statistics. A new and simple bandwidth selection technique is proposed for semiparametric inferences in the partially linear model. Numerical examples are presented to illustrate the proposed methods.;Normalization of microarray data is essential for removing experimental biases and revealing meaningful biological results. Motivated by a problem of normalizing microarray data, Fan et al. (2003) propose a Semi-linear In-slide Model (SLIM). Part II demonstrates that this semiparametric model has a number of interesting features: the parametric component and the nonparametric component that are of primary interest can be consistently estimated, the former possessing a parametric rate and the latter having a nonparametric rate, while the nuisance parameters can not be consistently estimated. This is an interesting extension of the partial consistent phenomena observed by Neyman and Scott (1948), which itself is of theoretical interest. To aggregate information from other arrays, SLIM is generalized to account for across-array information, resulting in an even more dynamic semiparametric regression model. The asymptotic normality for the parametric components and the rate of convergence for the nonparametric component are established. The results are augmented by simulation studies.
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