摘要
小波估计方法一直是统计学领域中的研究热点和难点问题,在数据压缩、流体湍流、信号和图像处理、地震勘探等领域有着广泛的应用价值.本文以小波估计方法在数理统计中的应用为研究对象,重点介绍小波估计方法的基本理论、门限函数种类,以及小波估计方法在完全数据、不完全数据和纵向数据下的研究成果.由于数据的复杂性和不完全性,导致传统的研究方法不再适用,需要结合左截断数据、右删失数据、缺失数据和纵向数据的特点,利用插入法、回归校正法、插补法和可逆概率加权法,构造被估函数的非线性小波估计量,研究非线性小波估计量平均积分二次误差(meanintegral square error,MISE)的渐近展开式和估计量的渐近正态性;讨论被估函数存在有限个不连续点时,非线性小波估计量MISE仍然成立;证明非线性小波估计量在包含很多不连续函数的Besov空间里的一致收敛性;利用小波估计方法研究回归模型中参数和非参数估计量的相合性和收敛速度;最后简要探讨小波估计方法未来的可能发展方向.
Wavelet estimation method has always been one hot and difficult problem in Statistics,and has wide application value in data compression,turbulence analysis,image and signal processing and seismic exploration,etc.The research object of this paper is the application of wavelet estimation method in mathematical statistics,focuses on the basic theory of wavelet estimation method,the types of threshold,and research achievements of the wavelet estimation method under complete data,incomplete data and longitudinal data.Due to the complexity and incompleteness of the data,traditional research methods are no longer applicable.It needs to combine with the characteristics of left truncated data,right censored data,missing data and longitudinal data,use the plug-in,calibration regression,imputation and inverse probability weighting methods.The nonlinear wavelet estimators of estimated functions are constructed,study the asymptotic expansion for mean integral square error(MISE)of nonlinear wavelet estimators and prove the asymptotic normality of estimators.The asymptotic expansions of MISE are still true for the estimated function with finite discontinuous points,and verify the uniform convergence rate of nonlinear wavelet estimators in Besov spaces,which contain unsmoothed functions;as well the wavelet method is used to study the consistency and convergence rate of the parametric and nonparametric parts for the semi-parametric regression models.Finally,the potential development direction of wavelet method is briefly discussed.
作者
邹玉叶
范国良
ZOU Yuye;FAN Guoliang(Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE,School of Statistics,East China Normal University,Shanghai,200062,China;College of Economics and Management,Shanghai Maritime University,Shanghai,201306,China)
出处
《应用概率统计》
CSCD
北大核心
2021年第2期201-220,共20页
Chinese Journal of Applied Probability and Statistics
基金
中国博士后科学基金项目(批准号:2019M651422)资助。
关键词
渐近性质
不完全数据
线性小波估计
纵向数据
非线性小波估计
asymptotic properties
incomplete data
linear wavelet estimation
longitudinal data
nonlinear wavelet estimation
