基于离差平方和分解式的线性回归检验统计量的构建  

Construction of Linear Regression Test Statistics Based on the Sum of Squared Deviations Decomposition Formula

作　　者：赵云[1] 张虹[1] ZHAO Yun;ZHANG Hong(School of Mathematical Sciences,Gansu Minzu Normal University,Hezuo Gansu 747000)

机构地区：[1]甘肃民族师范学院数学科学学院,甘肃合作747000

出　　处：《甘肃高师学报》2024年第5期1-6,共6页Journal of Gansu Normal Colleges

基　　金：2022年甘肃省高校创新创业教育教学改革研究项目“以创新创业教育为导向的《概率论与数理统计》课程实践教学研究”;2024年甘肃民族师范学院教学成果培育重点项目“大数据背景下数理统计课程‘四路径,四层次’实践教学模式研究”。

摘　　要：文章基于样本数据离差平方和分解式,分析影响变量之间相关关系的因素.将离差平方和分解成回归平方和与残差平方和两部分,呈现出数据变异的根源在于变量自身取值的差异性和存在不可控制的随机因素两方面叠加的结果.建立线性回归数学模型,应用最小二乘估计法先求出回归方程参数的估计值,由估计值的数学期望、方差的性质,得到回归平方和、残差平方和服从正态分布、χ2分布的结论,再依据F分布、t分布的结构模式,构造回归分析显著性检验的F统计量、t统计量.文章理清了从离差平方和分解式、回归分析模型到选择检验统计量的思维线索.Based on the sample data of the sum of squared deviations,this paper analyzes the factors that affect the correlation between variables,and decomposes the square sum of deviation into two parts:the square sum of regression and the square sum of residual,showing that the root of data variation lies in the difference of variables'own values and the existence of uncontrollable random factors.The least square estimation method is used to get the estimated values of the parameters of the regression equation.According to the mathematical expectation of the estimated values and the nature of variance,it is concluded that the sum of squares of regression and the sum of squares of residuals obey normal distribution andχ2 distribution.Then,according to the structural model of F distribution and t distribution,the F statistics and t statistics of regression analysis significance test are constructed.The article clarifies the thinking clues from the decomposition formula of sum of squared deviations,regression analysis model to the selection of test statistics.

关 键 词：离差平方和分解式 线性回归 显著性检验 统计量的构建 

分 类 号：O212[理学—概率论与数理统计]