Question

Let \( X_1, X_2, ...., X_{10} \) be a random sample of size 10 from a population having \( N(0, \theta^2) \) distribution, where \( \theta>0 \) is an unknown parameter. Let \( T = \frac{1}{10} \sum_{i=1}^{10} X_i^2 \). If the mean square error of \( cT \) (for \( c>0 \)), as an estimator of \( \theta^2 \), is minimized at \( c = c_0 \), then the value of \( c_0 \) equals

• A. \( \frac{5}{6} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{1}{2} \)

Hint:

- The mean square error (MSE) is a key measure in the performance of an estimator. It is the sum of the variance and the square of the bias.
- To minimize MSE, we differentiate the MSE expression with respect to the parameter and find the value that minimizes it.

Answer 1

The Correct Option is A

The mean square error (MSE) for an estimator is the sum of the variance and the square of the bias. For the estimator \( cT \), we want to minimize the MSE with respect to the constant \( c \). Using the properties of the variance of \( T \) and the bias of \( cT \), we minimize the MSE expression. The optimal value of \( c \) that minimizes the MSE is \( c_0 = \frac{5}{6} \).
The correct answer is (A) \( \frac{5}{6} \).