# 统计计算|Statistical Computing代写 STAT 535

Population regression function, or simply, the regression function:

For the Normal linear model
$$\mathrm{E}\left(Y_{i}\right)=\mu_{i}=\mathbf{x}{i}^{T} \boldsymbol{\beta} ; \quad Y{i} \sim \mathrm{N}\left(\mu_{i}, \sigma^{2}\right)$$
for independent random variables $Y_{1}, \ldots, Y_{N}$, the deviance is
$$D=\frac{1}{\sigma^{2}} \sum_{i=1}^{N}\left(y_{i}-\widehat{\mu}_{i}\right)^{2}$$

$$D_{0}=\frac{1}{\sigma^{2}} \sum_{i=1}^{N}\left[y_{i}-\widehat{\mu}{i}(0)\right]^{2}$$ and $$D{1}=\frac{1}{\sigma^{2}} \sum_{i=1}^{N}\left[y_{i}-\widehat{\mu}_{i}(1)\right]^{2} .$$

## STAT535 COURSE NOTES ：

If $\mathrm{E}(\mathbf{y})=\mathbf{X} \boldsymbol{\beta}$ and $\mathrm{E}\left[(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})^{T}\right]=\mathbf{V}$, where $\mathbf{V}$ is known, we can obtain the least squares estimator $\tilde{\beta}$ of $\beta$ without making any further assumptions about the distribution of $\mathbf{y}$. We minimize
$$S_{w}=(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})^{T} \mathbf{V}^{-1}(\mathbf{y}-\mathbf{X}) \boldsymbol{\beta}$$
The solution of
$$\frac{\partial S_{w}}{\partial \beta}=-2 \mathbf{X}^{T} \mathbf{V}^{-1}(\mathbf{y}-\mathbf{X} \beta)=0$$
is
$$\tilde{\boldsymbol{\beta}}=\left(\mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{X}\right)^{-1} \mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{y}$$