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====== Quiz 4 ======

===== Exercise 1: hat matrix =====

Consider a multiple linear regression model $Y=X\beta +\epsilon$ where $\mathrm{rank}(X)=p$ and $\PE[\epsilon]=0$ and $\Var(\epsilon)=\sigma^2 I$ are satisfied. We write $X=[X_1|\ldots|X_p]$. 
Denote by $\hat \beta$ the OLS estimator of $\beta$ obtained from the regression of $Y$ on $\{X\beta,\ \beta \in \rset^p\}$. Define $P_X=X(X'X)^{-1}X'=[h_{i,j}]_{1\leq i, j\leq n}$. 
 Which of the following statements are true? 
 
  * 1. Assume that $n=3$, the situation where we have $(h^2_{12},h^2_{13})=(0.2,0.2)$ is impossible. **TRUE** 
  * 2. The situation where $h_{ii}=0.5$, and for all $j\neq i$, $h_{ij}=0$ may happen sometimes. **FALSE**
  * 3. If $\epsilon \sim N(0,\sigma^2 I_n)$ and if the (external) studentized residual $t_i^*$ satisfies $t_i^*<-2$, then we consider that $(x'_i,y_i)$ is a leverage point. **FALSE**

===== Exercise 2: Hoeffding's inequality =====

Let $(X_i)_{1\leq i \leq n}$ be iid random variables such that $\PP(X_1 \in [0,1])=1$. Which of the following statements are true? 

  * 4. According to Hoeffding's inequality, $\PP\lr{\left|\frac{\sum_{i=1}^n X_i}{n} - \PE[X]\right| > \epsilon} \leq 2 \rme^{-2 \epsilon^2/n}$ **FALSE**
  * 5. According to Hoeffding's inequality, $\PP\lr{\left|\frac{\sum_{i=1}^n X_i}{n} - \PE[X]\right| > \epsilon} \leq 2 \rme^{-2 n^2\epsilon^2}$ **FALSE**
  * 6. According to Hoeffding's inequality, $\PP\lr{\left|\sum_{i=1}^n (X_i - \PE[X])\right| > \epsilon} \leq 2 \rme^{-2 n\epsilon^2}$ **FALSE**