函数型部分线性乘积模型的B-样条估计  

B-spline Estimations of Partially Functional Linear Multiplicative Model

作　　者：丁建华 余平[2] 丁燕萍 DING JIANHUA;YU PING;DING YANPING(School of Mathematics and Statistics,Shanci Datong University,Datong 037009,China;School of Mathematics and Computer Sciences,Shanci Normal University,Taiyuan 030031,China)

机构地区：[1]山西大同大学数学与统计学院,大同037009 [2]山西师范大学数学与计算机科学学院,太原030031

出　　处：《应用数学学报》2024年第2期204-225,共22页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(批准号:12071267);全国统计科学研究项目(批准号:2022LY089);山西省自然科学基金(批准号:20210302124262,20210302124530,202203021222223);山西省高等学校教学改革创新项目(批准号:J20220450,J2021512);山西师范大学自然科学基金(批准号:JYCJ2022004)资助项目。

摘　　要：本文研究了函数型部分线性乘积模型,该模型可用于响应变量为正数的函数型数据的统计建模问题,经过对数变换后模型转化为函数型部分线性模型.基于B-样条,通过极小化最小一乘相对误差(LARE)和最小乘积相对误差(LPRE),分别给出模型的LARE估计和LPRE估计,其中B-样条基的维数利用Schwarz信息准则选取.对两种估计方法分别给出斜率函数估计的相合性和参数部分估计的渐近正态性,并且证明了斜率函数的收敛率达到了非参数函数估计的最优速率.蒙特卡洛模拟用来比较所提出的方法与最小一乘(LAD)估计和最小二乘(LS)估计在不同误差分布下的有限样本性质,模拟结果表明所提方法是有效和实用的.最后通过一个实际数据分析的例子来说明模型的应用.In this paper,the partial functional linear multiplicative model is con-sidered.This model,which becomes partial functional linear regression after taking logarithmic transformation,is useful in analyzing data with positive responses.Based on B-splines,two estimation methods are proposed by minimizing the least absolute relative error(LARE)and the least product relative error(LPRE),respectively.The dimension of the B-spline bases is selected using the Schwarz information criteria.Con-sistency and asymptotic normality of the two methods are investigated.For the slope function,we prove that its convergence rate achieves the optimal rate of nonparametric function.Monte Carlo simulations are conducted to evaluate and compare the finite sample performance of the proposed estimators with the least squares(LS)estimator and the least absolute deviation(LAD)estimator under the different random error settings.Simulation results show that the proposed methods are comparable to other methods.Finally,an example of real data analysis is given to illustrate the application of the model.

关 键 词：函数型数据 乘积模型 B-样条 渐近性质 

分 类 号：O212.7[理学—概率论与数理统计]