Stata：缺失值与多重补漏-misstable-D204

发布时间：2021-07-02 阅读 192

\section{附录 A: 二元正态分布}%Bivariate Normal Distribution %If the joint density function of two random variables $(u_{1}, u_{2})$ has the form 如果随机变量 $u_{1}$ 和 $u_{2}$ 具有二元标准正态分布，则 $(u_{1}, u_{2})$ 的联合密度函数为：

\begin{equation}\label{eq:limdep-Winkel-7-76} f\left(u_{1}, u_{2}\right)=\frac{1}{2 \pi} \frac{1}{\sqrt{1-\rho^{2}}} \exp \left{-\frac{1}{2}\left(\frac{u_{1}^{2}+u_{2}^{2}-2 \rho u_{1} u_{2}}{1-\rho^{2}}\right)\right} \end{equation}

u_{1}

and

u_{2}

are said to have a bivariate standard normal distribution. The marginal distributions are standard normal:

u_{1} \sim N (0, 1)

and

u_{2} \sim

Normal

(0, 1)

. The correlation between

u_{1}

and

u_{2}

is given by

ρ .

The basic rule for conditional densities is that

f (u_{1} ∣ u_{2}) = f (u_{1}, u_{2}) / f (u_{2}) .

Since 它们的边际分布是标准正态分布：

u_{1} \sim N (0, 1)

和

u_{2} \sim N (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： $f (u_{1} ∣ u_{2}) = f (u_{1}, u_{2}) / f (u_{2})$ 。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 $f (u_{1}, u_{2}) / f (u_{2})$ 可以写成 %and using some simplifications, we find that the ratio $f (u_{1}, u_{2}) / f (u_{2})$ can be written as

\begin{equation}\label{eq:limdep-Winkel-7-78} f\left(u_{1} \mid u_{2}\right)=\frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{1-\rho^{2}}} \exp \left{-\frac{1}{2}\left(\frac{\left(u_{1}-\rho u_{2}\right)^{2}}{1-\rho^{2}}\right)\right} \end{equation}

%In other words,

u_{1} ∣ u_{2}

has a normal distribution as well, and

u_{1} ∣ u_{2} \sim N (ρ u_{2}, 1 - ρ^{2})

. 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)

a n d

z_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right)

, a n d

\left(z_{1}, z_{2}\right) \sim Bivariate Normal(\left.m_{1}, m_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho\right)

d e n o t e s t h e g e n e r a l b i v a r i a t e n o r m a l d i s t r i b u t i o n . F o r t h e c o n d i t i o n a l e x p e c t a t i o n, w e o b t a i n 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 \begin{aligned} E (z_{1} ∣ z_{2}) & = m_{1} + σ_{1} E (u_{1} ∣ z_{2}) & = m_{1} + σ_{1} ρ (\frac{z_{2} - m_{2}}{σ_{2}}) & = m_{1} + ρ \frac{σ_{1}}{σ_{2}} (z_{2} - m_{2}) & = m_{1} + \frac{σ_{12}}{σ_{2}^{2}} (z_{2} - m_{2}) \end{aligned} T h i s r e s u l t s h o w s t h a t u n d e r b i v a r i a t e n o r m a l i t y, i t i s t h e c a s e t h a t

\mathrm{E}\left(z_{1} \mid z_{2}\right)=

\alpha+\beta z_{2}

, w h e r e

\beta=\sigma_{12} / \sigma_{2}^{2}

a n d

\alpha=m_{1}-\beta m_{2}$. Thus, the conditional expectation function of the bivariate normal distribution is a linear function, or a "linear regression".

Stata：缺失值与多重补漏-misstable-D204

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