Question

Let \( X_1, X_2, \dots, X_n \) be a random sample from a continuous distribution with the probability density function for \( \lambda>0 \) 

 To test the hypothesis \( H_0: \lambda = \frac{1}{2} \) against \( H_1: \lambda = \frac{3}{4} \) at the level \( \alpha \) (with \( 0<\alpha<1 \)), which of the following statements is (are) TRUE? 
 

• A. The most powerful test exists for each value of \( \alpha \)
• B. The most powerful test does not exist for some values of \( \alpha \)
• C. If the most powerful test exists, it is of the form: Reject \( H_0 \) if \( X_1^2 + X_2^2 + \dots + X_n^2>c \) for some \( c>0 \)
• D. If the most powerful test exists, it is of the form: Reject \( H_0 \) if \( X_1^2 + X_2^2 + \dots + X_n^2 \geq c \) for some \( c \geq 0 \)

Hint:

The Neyman-Pearson Lemma provides a method for finding the most powerful test for simple hypotheses. This involves using the likelihood ratio, which often leads to tests involving the sum of squares of the observations.

Answer 1

The Correct Option is A, C

Step 1: Understand the hypothesis testing. 
The most powerful test for simple hypotheses is given by the Neyman-Pearson Lemma, which states that the likelihood ratio test is the most powerful test. In this case, the likelihood ratio test will involve the sum of squares of the observations, because the distribution of \( X^2 \) is relevant to the hypothesis. 
Step 2: Analyzing the options. 
(A) The most powerful test exists for each value of \( \alpha \): This is not true. The most powerful test does not always exist for all values of \( \alpha \). 
(B) The most powerful test does not exist for some values of \( \alpha \): This is true. A most powerful test may not exist for every \( \alpha \). 
(C) If the most powerful test exists, it is of the form: Reject \( H_0 \) if \( X_1^2 + X_2^2 + \dots + X_n^2>c \) for some \( c>0 \): This is correct. The test statistic will involve the sum of squares, as this is related to the likelihood ratio for \( \lambda \). 
(D) If the most powerful test exists, it is of the form: Reject \( H_0 \) if \( X_1^2 + X_2^2 + \dots + X_n^2 \geq c \) for some \( c \geq 0 \): This is incorrect. The test involves a strict inequality for the critical region. 
Step 3: Conclusion. 
The correct answer is C, as the most powerful test involves rejecting \( H_0 \) if the sum of squares exceeds a threshold.