Asymptotic Distribution Of Nonparametric Kernel Estimation For Functional Stationary Ergodic Data

In recent years,with the progress of computer and communications technology,we can find and collect a series of functional data,for example, some meteorological data, economic indicators,and so on,these data are widely applied in many fields such as m-edicine, economics, environment among others.Therefore, many experts at home and abroad are interested in the statistical research of functional data.Because traditional data analysis method is limited to the study of the conventional static data, the functi-onal data analysis methods are mainly convert the observed discrete data into a funct-ion, and allows us to estimate the value of all the arguments on a range, thus one can obtain more accurate information. The most common method of functional data anal-ysis is to combine the functional data and non-parametric statistics,in addition, the no-nparametric time series analysis has received considerable attention in non-parametric statistical inference, in practical problems, with respect to α-mixing time series, it is more easier to verify the ergodic of a nonlinear time series, therefore, it is necessary to research the properties of non-parametric estimator for functional stationary ergodi-c data.This article aims to consider the asymptotic normality of modified kernel regressio-n and conditional density estimation when the explanatory variable X is characteristic and the response variable Y values in real space.In this paper,we construct the modif-ied kernel estimation of the regression function r (x) and the kernel estimation of co-nditional density based on the functional stationary ergodic data by N-W kernel esti-mator method, and research the asymptotic normality of modified kernel regression a-nd conditional density estimation. After the study of asymptotic distribution of modif-ied kernel regression and conditional density estimator, we can get the confidence re-gion, thus we can obtain the relatively good estimator area,so that the results of this paper has more practical value, which extend the a-mixing data to the functional stati-ionary ergodic data.