利用Cauchy-Schwarz不等式估计回归系数

Use Cauchy-Schwarz Inequality Estimated Regression Coefficient

根据随机变量情形下的Cauchy-Schwarz不等式,推广得到关于二阶矩的推论:随机变量X、Y的方差均存在且不为零,如果随机变量X、Y依概率1具有线性关系P{Y=aX+b}=1,则a=E(X-EX)(Y-EY)/E(X-EX)2,b=EY-aEX,同时给出线性回归问题中回归系数的一种矩估计法。

From the Cauchy-Schwarz inequality in form of random variable, two results of Second-order moment are obtained ： Variances of random variable X, Y exists and not 0, if the probability of X, Y has linear relationship P { Y =aX＋b} = 1 is 1, then a=E（X-EX）（Y-EY）/E（X-EX）^2, b = EY - aEX. Furthermore a method of estimating regression coefficient is also obtained.