Research Based On Asymptotic Properties Of Regression Function And Conditional Density In The Single Functional Index Model

In practice, with the development of technology and method for obtaining data,the observed data that more and more research fields collected has functional cha-racteristic. For this reason, functional data about theories is one of the hot topics instudies of the modern statistics. Meanwhile, great challenge has been brought onthe statistical inference and data analysis because of functional data is differentfrom the traditional acquired that is infinite dimension, so it is necessary to reducethe dimensionality before constructing data model. However, the single-indexmodel is an important semi-parametric method by transforming multivariate vectorsinto univariate index parameters, which can also capture the important features inhigh dimensional data. Therefore, the research focused on single functional indexmodel has been gradually paid more attention.In this thesis, we mainly study the asymptotic behaviors of regression functionand conditional density function in the single functional index model for functionaldata, and obtain some better results. The details are given as follows:Firstly, based on-mixing dependence of functional time series data, weachieve almost uniform complete convergence rate of nonparametric regressionfunction estimator by the Kolmogorov’s entropy.Secondly, we investigate the double kernel estimation of condition density ofcondition density function based on the single-index model for functional time se-ries data. The asymptotic normality of the conditional density estimator and theconditional mode estimator for the α-mixing dependence functional time seriesdata are successfully obtained, respectively.Thirdly, as applications, the asymptotic1-cξonfidence interval of the condi-tional density function and the conditional mode are also presented.