Stata:缺失值与多重补漏-misstable-D204

发布时间:2021-07-02 阅读 778

\section{附录 A: 二元正态分布}%Bivariate Normal Distribution %If the joint density function of two random variables (u1,u2) has the form 如果随机变量 u1 和 u2 具有二元标准正态分布,则 (u1,u2) 的联合密度函数为:

%u1 and u2 are said to have a bivariate standard normal distribution. The marginal distributions are standard normal: u1N(0,1) and u2 Normal (0,1). The correlation between u1 and u2 is given by ρ. The basic rule for conditional densities is that f(u1u2)=f(u1,u2)/f(u2). Since 它们的边际分布是标准正态分布:u1N(0,1) 和 u2N(0,1)。二者的相关性由参数 ρ 决定。

我们通常更为关心条件密度函数:f(u1u2)=f(u1,u2)/f(u2)。由于 \beginequation*}\label{eq:limdep-Winkel-7-77} f\left(u_{2}\right)=\frac{1}{\sqrt{2 \pi}} \exp \left{-\frac{1}{2} u_{2}^{2}\right} \end{equation*} 再做一些简化处理,可知 f(u1,u2)/f(u2) 可以写成 %and using some simplifications, we find that the ratio f(u1,u2)/f(u2) can be written as

%In other words, u1u2 has a normal distribution as well, and u1u2N(ρu2,1ρ2). The conditional expectation function of the standard bivariate normal distribution is given by 可见,$u_{1}u_{2也服从正态分布,即u_{1} | u_{2} \sim N\left(\rho u_{2}, 1-\rho^{2}\right)。由此可得标准二元正态分布的的条件期望函数为: \begin{equation}\label{eq:limdep-Winkel-7-79} \mathrm{E}\left(u_{1} \mid u_{2}\right)=\rho u_{2} \end{equation} %The results can be generalized to allow for marginal normal distributions with non-zero means and non-unit variances. Let 进一步考虑“非零均值”和“非单位方差”的一般化情形: \begin{equation}\label{eq:limdep-Winkel-7-80} \begin{aligned} &z_{1}=m_{1}+\sigma_{1} u_{1} \ &z_{2}=m_{2}+\sigma_{2} u_{2} \end{aligned} \end{equation} %so that z_{1} \sim N\left(m_{1}, \sigma_{1}^{2}\right)andz_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right),and\left(z_{1}, z_{2}\right) \sim Bivariate Normal(\left.m_{1}, m_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho\right)denotesthegeneralbivariatenormaldistribution.Fortheconditionalexpectation,weobtain简记为z_{1} \sim N\left(m_{1}, \sigma_{1}^{2}\right)z_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right),以及\left(z_{1}, z_{2}\right) \sim N(\left.m_{1}, m_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho\right)表示一般二元正态分布。对于条件期望,我们得到E(z1z2)=m1+σ1E(u1z2) =m1+σ1ρ(z2m2σ2) =m1+ρσ1σ2(z2m2) =m1+σ12σ22(z2m2)Thisresultshowsthatunderbivariatenormality,itisthecasethat\mathrm{E}\left(z_{1} \mid z_{2}\right)=\alpha+\beta z_{2},where\beta=\sigma_{12} / \sigma_{2}^{2}and\alpha=m_{1}-\beta m_{2}$. Thus, the conditional expectation function of the bivariate normal distribution is a linear function, or a "linear regression".