| Nonlinear time series analysis has gained much attention in recent years due primarily to the fact linear time series models have encountered various limitations in real applications and the development in nonparametric regression has established a solid foundation for nonlinear time series analysis. For example, the effect of technology on the economic growth, volatility of exchange returns, which follow nonlinear instead of simple linear prediction formulas. Effective tools for extracting information from such complex regression data have to be nonparametric in nature.;A smooth kernel estimator is proposed for multivariate cumulative distribution function in Chapter 2, extending the work on Yamato (1973) on univariate distribution function estimation. Under assumptions of strict stationarity and geometrically strong mixing, we establish that the proposed estimator follows the same pointwise asymptotically normal distribution of the empirical cdf, while the new estimator is a smooth instead of a step function as the empirical cdf. We also show that under stronger assumptions the smooth kernel estimator has asymptotically smaller mean integrated squared error than the empirical cdf, and converges to the true cdf uniformly almost surely at a rate of (n -1/2 log n). Simulated examples are provided to illustrate the theoretical properties. Using the smooth estimator, survival curves are given for real data applications.;"Curse of dimensionality" is a significant obstacle in high dimensional time series analysis, see Fan and Yao (2003). Several high dimensional data analysis techniques have been proposed to deal with this problem and Xia, Tong, Li and Zhu (2002) pointed out that there are essentially two approaches: function approximation and dimension reduction. GARCH model, Additive Coefficient Model (ACM) and Generalized Additive model (GAM) are good examples to represent these two approaches.;In Chapter 3, a cubic spline regression procedure is proposed to estimate the unknowns in the semiparametric GARCH model that is intuitively appealing due to its simplicity, and as such, can be used by non experts. The theoretical properties of the procedure is the same as the kernel procedure in Yang (2006), and simulated and real data examples show that the numerical performance is also comparable to the kernel method. The new method is computationally much more efficient and very useful for analyzing financial time series data.;In Chapter 4, a spline-backfitted kernel estimator is proposed for estimating the unknown component functions malphal based on a geometrically strong mixing sample following model (1.3.1) under minimal smoothness assumptions. The idea is to employ one step backfitting after the spline pilot estimators, and then follow up with kernel smoothing, which combines the fast computing of polynomial spline smoothing and the good asymptotic property of kernel smoothing. Thus, the spline-backfitted kernel estimator is both computationally expedient for analyzing very high dimensional time series, and theoretically reliable to make inference on the component functions with confidence.;In Chapter 5, a spline-backfitted kernel (SBK) estimator is proposed for the Generalized Additive Model time series data with oracle efficiency. It is both computationally expedient and theoretically reliable, and simulation evidence strongly corroborates the asymptotic theory. |
