Some Applications Of Cubic Smoothing Spline

One primary application of Statistics is to seek an appropriate model based on observational data or sample data sets. There are many techniques to fit data sets, of which linear regression model might be the oldest classic regression model. In the beginning of the development of Statistics, linear regression model can fit data sets appropriately. Subsequently, it has been developed into generalized linear model because of some practical issues. The appearance of link function leads to establishment of generalized linear model. During the past one or two decade, the linear model and generalized linear model(GLM) has been extended to nonlinear model due to the complexity of interactions among variables. It is true that one can apply some specific function related to the factors of interest to fit the data set, provided the unspecified relationship among variables is known to some degree based on one's experience, which is called parametric model. When the underlying relationship among variables is completely unknown or partially known, nonparametric and semiparametric models are introduced. Nonparametric and semiparametric models have less constricted conditions on themselves and are more adaptive. Therefore, nonparametric and semiparametric model have become a hot topic in the past few years. Compared to that of the parametric model, the conditions required in the nonparametric and semiparametric model are relaxed further and more applicable in reality and so become more and more popular by people. As to nonparametric and semiparametric regression models, the most widely used regression methods include, spline method(such as smoothing spline, penalized spline and so on); local smoothness, such as Nadaraya-Watson kernel estimate, Gasser-Muller estimate, local polynomial method; the third is orthogonal series, including Fourier series and so on. Of all the spline methods, cubic smoothing spline is probably the most extensively used one in that it shares the simplicity of computation and appealing statistical properties. This dissertation plans to employ cubic smoothing spline to estimate both single-index parameter from single-index partially linear model and varying coefficients from varying coefficient partially linear model, including estimate method, estimate procedure and the related asymptotic properties. To illustrate the performance of our proposed method, we will present some simulations corresponding to the above two models, which includes some graphics such as histogram, box plot, Q-Q plot and estimated function curve.