Question

For $n \geq 2$, let $\epsilon_1, \epsilon_2, \ldots, \epsilon_n$ be i.i.d. random variables having the $N(0,1)$ distribution. Consider $n$ independent random variables $Y_1, Y_2, \ldots, Y_n$ defined by $Y_i = \beta + \epsilon_i$, $i = 1,2, \ldots, n$, where $\beta \in \mathbb{R}$. Define
$\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i$
$T_1 = \frac{2\bar{Y}}{n+1}$
$T_2 = \frac{1}{n} \sum_{i=1}^{n} \frac{Y_i}{i}$
Then which of the following statements is NOT correct?

• A. $T_1$ is an unbiased estimator of $\beta$
• B. $T_2$ is an unbiased estimator of $\beta$
• C. $\mathrm{Var}(T_1) < \mathrm{Var}(T_2)$
• D. $\mathrm{Var}(T_1) = \mathrm{Var}(T_2)$

Answer 1

Answer: (D)

$\mathrm{Var}(T_1) = \mathrm{Var}(T_2)$

Answer 2

The correct option is (D): $\mathrm{Var}(T_1) = \mathrm{Var}(T_2)$