这是一份umass麻省大学 STAT 535作业代写的成功案例
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
$$
\tilde{\boldsymbol{\beta}}=\left(\mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{X}\right)^{-1} \mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{y}
$$