统计研究 ›› 2018, Vol. 35 ›› Issue (6): 117-128.doi: 10.19343/j.cnki.11-1302/c.2018.06.012

• • 上一篇    

相依函数型数据的局部回归估计的渐近正态性

李双博   

  • 出版日期:2018-06-25 发布日期:2018-06-22

Asymptotic Normality of Partially Regressive Estimators for Dependent Functional Data

Li Shuangbo   

  • Online:2018-06-25 Published:2018-06-22

摘要: 函数型数据研究近年来为越来越多的学者所重视,其在天文,医药,经济现象,生态环境及工业制造等诸多方面均有重要应用.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重要常用方法.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. Marin, Ferraty, Vieu [Journal of Nonparametric Statistics, 22 (5) (2010), pp.617-632] 提出了非参函数型模型的局部回归估计. 这种估计可以看作是局部多项式估计在函数型数据背景下的一个推广.这种方法提出后,许多学者进一步研究了这种方法,考察了这种方法的收敛速度和极限分布,并将这种方法应用到不同的模型中以适应实际需求.但是,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,我们采用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.此外我们还采用数据模拟的方法进一步验证了渐近正态的结果.

关键词: 函数型数据, 局部回归估计, 渐近正态性

Abstract: 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.

Key words: Functional data, Partially Regressive Estimator, Asymptotic Normality