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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (1): 107-119.doi: 10.3969/j.issn.1005-3085.2022.01.008

• • 上一篇    下一篇

部分线性变量含误差的网格数据模型的估计

马俊玲1,    王立春2   

  1. 1. 上海财经大学统计与管理学院，上海 200433
    2. 北京交通大学数学与统计学院，北京 100044
  • 出版日期:2022-02-15 发布日期:2022-04-15
  • 基金资助:
    国家自然科学基金 (11371051)；教育部人文社会科学规划基金 (21XJJA910001).

A Partially Linear Errors-in-Variables Model with Observations at Grid Points

MA Junling1,   WANG Lichun2   

  1. 1. School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433 
    2. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044
  • Online:2022-02-15 Published:2022-04-15
  • Supported by:
    The National Natural Science Foundation of China (11371051); the MOE Project of Humanities and Social Sciences (21XJJA910001).

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摘要：

针对网格数据构建了部分线性变量含误差模型。在部分线性变量含误差的网格数据模型解释变量为固定设计的情形下，分别在单向维度观测样本容量趋于无穷和两个维度观测样本容量同时趋于无穷时对模型线性部分的未知参数、误差方差和非线性部分的未知函数进行了大样本统计推断研究。对模型线性部分未知参数和误差方差的改进的最小二乘估计量，在较弱的条件下推出估计量具有强相合性并得到收敛速度；对模型非线性部分未知函数的样条函数估计量，在一定的条件下证明了其具有强相合性并得到其收敛速度。在一定的正则条件下，分别对单向维度观测样本容量趋于无穷和两个维度观测样本容量同时趋于无穷的两种情形证明了模型线性部分未知参数和误差方差的估计量服从渐近正态分布，并对每种情形给出了渐近方差的表达式。

关键词: 网格数据, 变量含误差, 部分线性模型, 渐近正态性

Abstract:

For grid data, partially linear errors-in-variables models are constructed and studied. Large sample statistical inferences are made for the models as sample sizes of both two dimensions and one dimension tend to infinity respectively. For the modified least square estimators of the unknown parameters of the linear part and the variance of model error, their strong consistencies and convergence rates are obtained under weak conditions. For the spline function estimator of the unknown function of the nonlinear part, its strong consistency and convergence rate are obtained under certain conditions. Further, for the estimators of the parameters of the linear part and the variance of model error, their asymptotical normality are proved under some regularity conditions as sample sizes of two dimensions and one dimension tend to infinity respectively, and the asymptotical variances are provided.

Key words: grid data, errors-in-variables, partially linear models, asymptotic normality

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