# 概率和随机过程 Probability and Stochastic Processes STAT4061

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Let $Z$ have the standard normal distribution, Let $V$ be 0 or 1 with probabilities $1-p, p$ independent of $Z$. Then
$$\varepsilon=\left[\sigma_{1}(1-V)+\sigma_{2} V\right] Z= \begin{cases}\sigma_{1} Z, & \text { if } \quad V=0 \ \sigma_{2} Z, & \text { if } \quad V=1\end{cases}$$
has the “contaminated normal distribution,” with c.d.f.
$$F(x)=(1-p) \Phi\left(x / \sigma_{1}\right)+p \Phi\left(x / \sigma_{2}\right)$$
$F$ has mean $E(\varepsilon)=0$, variance $\operatorname{Var}(\varepsilon)=(1-p) \sigma_{1}^{2}+p \sigma_{2}^{2}$. The density of $E$ is plotted.

Now consider simple linear regression $Y_{i}=\beta_{0}+\beta_{i} x_{i}+\varepsilon_{i}$ for $i=1, \ldots, n$, where the $\varepsilon_{i}$ ‘s are a random sample from $F$ above. Take $n=10, x_{i}=i$ for $i=1, \ldots, 9$ and $x_{10}=30$. Take $\sigma_{1}=2, \sigma_{2}=20, p=0.2$.

Thus, $d_{10,10}$ as defined in Eicher’s Theorem is $\left(\frac{1}{10}\right)+\frac{(30-7.5)^{2}}{622.5}=0.913$,

## STAT4061 COURSE NOTES ：

$$e_{i}=e_{i}\left(b_{0}, b_{1}\right)=y_{i}-\left(b_{0}+b_{1} x_{i}\right) \quad \text { and } \quad Q\left(b_{0}, b_{1}\right)=\sum \rho\left(e_{i}\left(b_{0}, b_{1}\right)\right)$$
Let
$$Q^{0}\left(b_{0}, b_{1}\right)=\frac{\partial}{\partial b_{0}} Q\left(b_{0}, b_{1}\right)=\sum \psi\left(e_{i}\right)$$
and
$$Q^{1}\left(b_{0}, b_{1}\right)=\frac{\partial}{\partial b_{1}} Q\left(b_{0}, b_{1}\right)=\sum \psi\left(e_{i}\right) x_{i}$$