Asymptotical Properties Of Modified Kernel Regression Estimation For Functional Time Series Data

In recent years, with the continuous improvement of information technology,and the rapid development of computer, we often collected a functional data,whichbe seen as a sample track of the continuous time random process, such data arewide range of applications in medicine, economics, environmentrics etc, such asatmospheric temperature curve data analysis, the human growth curve data analysis.Therefore, the nonparametric statistics for functional data is concerned by themajority of scholars, many authors have constructed the estimator of regressionfunction and obtained its large sample properties: on the one hand, such asconsistency and asymptotic distribution in the setting of finite dimension case whenthe data is i.i.d. or has some dependence. On the other hand, in the case of infinitedimension,with the development of the theory of functional data, many attentionshave been paid to asymptotical properties of regression function.In this dissertation, firstly, under dependent functional data, we investigate amodified kernel estimation of the regression function r (x)=E (Y|X=x), wherethe explanatory variables take values in some abstract semi-metric function spaceand the response Y is real-valued. The details are given as follows:Firstly, under α-mixing dependence functional data, we establish theasymptotic normality of this estimator by the central limit Theorem for α-mixingtriangular arrays. As an application, the asymptotic (1-α)confidence interval forthe regression function operator r is given in this case.Secondly, under α-mixing dependence functional data, we obtain consistencyof recursive modified kernel regression estimator for α-mixing dependencefunctional data. The work of this dissertation extends the existing related results.