相依函数型数据的局部回归估计的渐近正态性

Asymptotic Normality of Partially Regressive Estimators for Dependent Functional Data

非参数统计是统计研究的一个重要方面,其中核函数估计和局部多项式方法是这一类研究中的常用方法。函数型数据的非参数方法中以核函数估计方法较为常见,且其收敛速度与极限分布无论在独立情形还是相依情形都有理论结果。而局部多项式的研究在函数型数据背景下较为少见,原因在于将局部多项式方法推广到函数型数据背景一直是一个难题,前人的研究都要求数据具有独立同分布的性质,然而许多实际数据并不符合这一假设。本文研究了在相依函数型数据情形下局部回归估计的渐近正态性。由于估计方法有差异,核函数估计的研究方法无法直接推广到局部回归估计,而相依性结构也给研究带来了一些挑战,本文采用Bernstein分块方法将相依性问题转化为渐近独立的问题,从而得到了估计的渐近正态性,同时采用数据模拟的方法进一步验证了渐近正态的结果。

Nonparametric statistics is one important aspect of statistical research,in which the kernel estimation and partial polynomial methods are commonly used. It is quite common to use kernel estimation for functional data. In terms of its convergence rate and asymptotic distribution the theoretical conclusions have already made,no matter it is independent or dependent. It is quite rare to use partial polynomial estimation for functional data analysis,because it is always a conundrum to apply functional data to the partial polynomial estimation. The researches by our predecessors require the data used independent with identical distribution,which is contrary to much of the real data. This paper studies the asymptotic normality of partial regressive estimators for dependent functional data. The methodology of kernel function estimation cannot be extended directly to partially regressive estimation,moreover,the dependent structure also brings up some challenges to our research. This paper adopts the Bernstein Block method to convert the dependent issue into asymptotic independent so as to obtain the asymptotic normality of the estimators. In addition,a simulation study is done to further justify the result of asymptotic normality.