Modern technological advances have led to the explosion in the collection of complex data such as functional/longitudinal, nonlinear time series, and mixed data, and data with jumps.In recent years, there has been a growing interest in developing statistical tools to analyze these data primarily due to the fact that traditional parametric methods are unrealistic in applications. Non- and semi- parametric methods as alternatives have been widely recognized as powerful tools for complex data analysis, which relax the usual assumptions of parametric methods and enable us to explore the data more flexibly so as to uncover data structure that might otherwise be missed.This dissertation develops statistical theories and methods in spline regression for those complex data mentioned before, with applications to medical science, finance and economics.In Chapter 2, procedures to detect jumps in the regression function via constant and linear splines are proposed based on the maximal differences of the spline estimators among neighboring knots. Simulation experiments corroborate with the asymptotic theory, while the computing is extremely fast. The detecting-procedure is illustrated in analyzing the thickness of pennies data set.In Chapter 3, asymptotically simultaneous confidence bands are obtained for the mean function of the functional regression model, using piecewise constant spline estimation. Simulation experiments corroborate the asymptotic theory. The confidence band procedure is illustrated by analyzing CD4 cell counts of HIV infected patients.A spline-backfitted kernel smoothing method is proposed in Chapter 4 for partially linear additive autoregression model. Under assumptions of stationarity and geometric mixing, the proposed function and parameter estimators are oracally efficient and fast to compute. Simulation experiments confirm the asymptotic results. Application to the Boston housing data serves as a practical illustration of the method.Chapter 5 considers the problem of estimating a relationship nonparametrically using regression splines when there exist both continuous and categorical predictors. The resulting estimator possesses substantially better finite-sample performance than either its frequency based peer or cross-validated local linear kernel regression or even additive regression splines (when additivity does not hold). Theoretical underpinnings are provided and Monte Carlo simulations are undertaken to assess finite-sample behavior.
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