A VBA‐based Simulation for Teaching Simple Linear Regression

In spite of the name, simple linear regression presents a number of conceptual difficulties, particularly for introductory students. This article describes a simulation tool that provides a hands‐on method for illuminating the relationship between parameters and sample statistics.     

CLASSROOM NOTE: A Roll of the Dice: Illustrating the Central Limit Theorem

The sampling (probability) distribution for the sum of dice is presented graphically as a compelling demonstration of the Central Limit Theorem.     

Mini-research Experience in Simple Linear Regression

Within the undergraduate mathematics curriculum, the topic of simple least-squares linear regression is often first encountered in multi-variable calculus where the line of best fit is obtained by using partial derivatives to find the slope and y-intercept of the line that minimizes the residual sum of squares. A markedly different approach from linear algebra, which could also be introduced in multi-variable calculus, obtains the regression line by vector projection. The latter viewpoint offers elegant proof of the equation relating the total, explained and unexplained variations. Consideration of data with the same regression line and correlation opens the door for a “mini-research experience” (MRE). A sequel MRE gives rise to an open Research Experience for Undergraduates topic to analyze reflection sequences and a fundamental connection between complex analysis and regression analysis. A few general guidelines and basic goals for MREs are included for those whose main interest is in undergraduate research.     

线性约束下的异方差回归模型参数的广义最小二乘估计

运用线性约束下回归模型的参数估计和异方差回归模型的参数估计方法,给出线性约束下的异方差回归模型的参数估计公式,比较该模型的残差平方和与无约束异方差回归模型的残差平方和的大小,利用此结论对模型的进一步研究和应用具有一定的理论和实际价值。     

A Story‐based Simulation for Teaching Sampling Distributions

Statistical inference relies heavily on the concept of sampling distributions. However, sampling distributions are difficult to teach. We present a series of short animations that are story‐based, with associated assessments. We hope that our contribution can be useful as a tool to teach sampling distributions in the introductory statistics classroom.     

Model Comparison in Regression

The instructor can help begining students master the numerous results of regression analysis by using helpful notation and by providing a framework for organising regression outputs. For the latter, model comparison is a useful operating notion.     

基于多元线性回归肤色模型的蒙特卡洛模拟

根据皮肤组织各层吸收体的分布特征,采用多元线性回归法构建皮肤组织的七层模型。利用蒙特卡洛方法模拟无限窄光束、平圆光束和高斯光束照射下光在皮肤组织中的传输特性,比较不同肤色组织光子传输的差异。结果表明,深肤色皮肤组织穿透深度较浅,生发层吸收能量密度较大,衰减较快,光能流率较小;高斯光束能量较平圆光束集中,适合精确作用小范围病损。     

Turning Points in Random Permutations

The mean and variance of the number of turning points in random permutations are evaluated. These results are applied to a test of randomness of fluctuations.     

EV线性模型中参数估计

证明了EV（errors-in-variables）线性模型中调整的最小二乘估计的强收敛性和渐近正态性,通过模拟实验验证了结果的正确性．     

约束线性回归模型回归系数的广义条件岭估计

对齐次等式约束线性回归模型回归系数的岭估计进行推广,得出广义条件岭估计．证明了在一定的条件下,在均方误差矩阵、均方误差及广义均方误差意义下广义条件岭估计都优于约束最小二乘估计,并且证明了广义条件岭估计是回归系数的可容许估计．     

线性回归模型中一类式子的推导

线性回归中的最小二乘估计具有线性无偏估计类中方差最小的性质,而这个性质的证明较为复杂,许多文献都予以省略,复相关系数的定义更是直接给出,并未从统计意义上加以解释,回归方程显著性检验所给出的统计量往往是从约束最小二乘出发得到,本文利用简单方法来推导这些性质,并且导出检验统计量的分布,解释了复相关系数的统计意义,以及回归模型中不相关的推导.     

非齐次等式约束线性回归模型回归系数的广义条件岭估计

时非齐次等式约束线性回归模型回归系数提出广义条件岭估计.证明了在一定的条件下,在均方误差矩阵、均方误差及广义均方误差意义下,广义条件岭估计都优于约束最小二乘估计和狭义条件岭估计.     

What Divisor for the Sample Variance?

Simulation is used to show why the divisor in the sample variance is n, - 1. Choice of the divisor for the sample standard deviation is also discussed.     

NMAR机制下线性回归模型的贝叶斯分析

本文对正态线性模型中存在不可忽略缺失数据时,如何使用贝叶斯方法对参数进行估计进行了探讨。并给出了相应的后验分布,介绍了相关的抽样方法。     

虚拟变量在线性回归模型中的一种应用

若干特异点处的统计量容易导致回归方程拟合不足,本文引入虚拟变量（Dummy Variables)给出一种较为实用的解决途径。     

多元线性回归模型分析的应用——以汉中市中学教师薪金影响因素分析为例

考虑汉中市的中学教师薪金与他们的资历、性别、教育程度及培训情况等因素及其关连性.该文主要应用多元线性回归模型,来拟合多个影响因素对一个变量的影响.通过经济模型的建立、分析及MATLAB程序求解加深对多元回归模型的理解.最后通过统计分析,说明此模型在实践中的可行性,并对短期内的情况进行预测.