这是一份nottingham诺丁汉大学MATH2010作业代写的成功案例

Let $Y_{i}=\beta x_{i}+\varepsilon_{i}, i=1,2,3$, suppose the $\left(x_{i}, Y_{i}\right)$ pairs $(1,1),(2,3),(3,7)$ are observed.
(a) Find the bootstrap estimate of the distribution of $\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}$. (Since the residual vector will not have components summing to zero, the components of
(b) Use the result of (a) to find an approximate $80 \%$ confidence interval on $\beta$.
(c) The permutation bootstrap method for regression analysis chooses the elements of $e^{*}$ without rather than with replacement, after correcting so the components sum to zero. Use this method to estimate the distribution of $\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}$.

MATH2010 COURSE NOTES ：

Suppose we want confidence intervals on a fixed number of linear combinations
$$
\eta_{1}=\mathbf{c}{1}^{\prime} \boldsymbol{\beta}, \quad \eta{2}=\mathbf{c}{2}^{\prime} \boldsymbol{\beta}, \ldots, \eta{r}=\mathbf{c}{r}^{\prime} \boldsymbol{\beta} $$ of the parameters of the linear model. We already have a technique for finding a $100(1-x)^{\circ}$ confidence interval on each such linear combination, namely $$ I{j}=\left[\hat{\eta}{j} \pm t{1-x_{j} / 2} S\left(\hat{\eta}{j}\right)\right] \quad \text { for } \quad j=1, \ldots, r, $$ where $\hat{\eta}{j}=\mathbf{c}{j}^{\prime} \hat{\boldsymbol{\beta}}, S^{2}\left(\hat{\eta}{j}\right)=S^{2} \mathbf{c}{j}^{\prime}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{c}{j}$ and $t_{1-x_{j}, 2}$ has $(n-k)$ d.f..
Let $E_{j}$ be the event that the confidence interval $I_{j}$ covers $\eta_{j}=\mathbf{c}{j}^{\prime} \beta$. We can make use of the Bonferroni Inequality to put a lower bound on the probability that all confidence intervals hold simultaneously. Thus $$ P\left(\bigcap{j=1}^{r} E_{j}\right)=1-P\left(\bigcup_{j=1}^{r} \bar{E}{j}\right) \geq 1-\sum{j=1}^{r} P\left(\bar{E}{j}\right)=1-\sum{1}^{r} \alpha_{j}
$$
If we want $P\left(\bigcap_{j=1}^{r} E_{j}\right) \geq 1-\alpha$ we therefore need only choose $x_{1}, \ldots, \alpha_{r}$ such that $\sum_{i}^{r} x_{j} \leq x$. The usual choice is $x_{j}=\alpha / r$.