Question

Let \( X_1, X_2 \) be a random sample from \( N(\theta, 1) \) distribution, where \( \theta \in \mathbb{R} \). Consider testing \( H_0: \theta = 0 \) against \( H_1: \theta \neq 0 \). Let \( \phi(X_1, X_2) \) be the likelihood ratio test of size 0.05 for testing \( H_0 \) against \( H_1 \). Then which one of the following options is correct?

• A. \( \phi(X_1, X_2) \) is a uniformly most powerful test of size 0.05
• B. \( E_{\theta}(\phi(X_1, X_2)) \geq 0.05 \, \forall \, \theta \in \mathbb{R} \)
• C. There exists a uniformly most powerful test of size 0.05
• D. \( E_{\theta = 0}(X_1 \phi(X_1, X_2)) = 0.05 \)

Hint:

The likelihood ratio test is designed to ensure the size of the test does not exceed the specified level. In this case, \( E_{\theta}(\varphi(X_1, X_2)) \geq 0.05 \) guarantees that the test's size is properly controlled.

Answer 1

The Correct Option is B

The likelihood ratio test is designed to control the size of the test, ensuring that the probability of a type I error (rejecting \( H_0 \) when \( \theta = 0 \)) is no greater than 0.05. Therefore, \( E_{\theta}(\phi(X_1, X_2)) \) should be at least 0.05 for all \( \theta \in \mathbb{R} \), ensuring the test maintains its size. Thus, option (B) is correct.