Statistical inference in time series analysis has been an important subject in various fields including climate science, economics, finance and industrial engineering among others. Numerous problems of research interest include statistical inference about unknown quantities, assessing structural stability and forecasting. These problems have been widely studied in the literature, but mainly for independent data, while in many applications involving time series data dependence is not unusual and in fact quite common. In this thesis, we incorporate serial dependence into the analysis by involving self-normalization in time series analysis. We start with the problem of testing whether there are change-points in a given time series. The method we propose does not require the number of change-points to be predefined, and thus is unsupervised. It does not require any tuning parameters and can be applied to a wide class to quantities of interest. The asymptotic distribution of the test statistic is studied and an approximation scheme is proposed to reduce testing procedure complexity. We then consider the problem of construction of confidence intervals, for which the conventional self-normalizer exhibits certain degrees of asymmetry when applied to quantities other than the mean. The method we propose provides a time-symmetric generalization to the conventional self-normalizer and leads to improved finite sample performance for quantities other than the mean.

Doctor of Philosophy / Department of Statistics / Weixin Yao / This dissertation contains two projects that are related to varying coefficient models. The traditional least squares based kernel estimates of the varying coefficient model will lose some efficiency when the error distribution is not normal. In the first project, we propose a novel adaptive estimation method that can adapt to different error distributions and provide an efficient EM algorithm to implement the proposed estimation. The asymptotic properties of the resulting estimator is established. Both simulation studies and real data examples are used to illustrate the finite sample performance of the new estimation procedure. The numerical results show that the gain of the adaptive procedure over the least squares estimation can be quite substantial for non-Gaussian errors. In the second project, we propose a unified inference for sparse and dense longitudinal data in time-varying coefficient models. The time-varying coefficient model is a special case of the varying coefficient model and is very useful in longitudinal/panel data analysis. A mixed-effects time-varying coefficient model is considered to account for the within subject correlation for longitudinal data. We show that when the kernel smoothing method is used to estimate the smooth functions in the time-varying coefficient model for sparse or dense longitudinal data, the asymptotic results of these two situations are essentially different. Therefore, a subjective choice between the sparse and dense cases may lead to wrong conclusions for statistical inference. In order to solve this problem, we establish a unified self-normalized central limit theorem, based on which a unified inference is proposed without deciding whether the data are sparse or dense. The effectiveness of the proposed unified inference is demonstrated through a simulation study and a real data application.