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?
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?
\[ \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i \]
\[ T_1 = \frac{2\bar{Y}}{n+1} \]
\[ T_2 = \frac{1}{n} \sum_{i=1}^{n} \frac{Y_i}{i} \]
Then which of the following statements is NOT correct?
Updated On: Oct 1, 2024
- \( T_1 \) is an unbiased estimator of \(\beta\)
- \( T_2 \) is an unbiased estimator of \(\beta\)
- \(\mathrm{Var}(T_1) < \mathrm{Var}(T_2)\)
- \(\mathrm{Var}(T_1) = \mathrm{Var}(T_2)\)
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The Correct Option is D
Solution and Explanation
The correct option is (D): \(\mathrm{Var}(T_1) = \mathrm{Var}(T_2)\)
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