Question

Let a random variable \( X \) follow a distribution with density \( f \in \{f_0, f_1\} \), where
\[ f_0(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise}, \end{cases} \] \[ f_1(x) = \begin{cases} 1 & \text{if } 1 \leq x \leq 2 \\ 0 & \text{otherwise}. \end{cases} \] Let \( \phi \) be a most powerful test of level 0.05 for testing \( H_0: f = f_0 \) against \( H_1: f = f_1 \) based on \( X \). Then which one of the following options is necessarily correct?

• A. \( E_{f_0}(\phi(X)) = 0.05 \)
• B. \( E_{f_1}(\phi(X)) = 1 \)
• C. \( P_f(\phi(X) = 1) = P_f(X>1), \forall f \in \{f_0, f_1\} \)
• D. \( P_{f_1}(\phi(X) = 1)<1 \)

Hint:

In a most powerful test, the test's power is maximized under the alternative hypothesis, which often results in a power of 1 under the alternative distribution.

Answer 1

The Correct Option is B

For the most powerful test, the power of the test should be maximized under the alternative hypothesis. Since the test is most powerful for \( f_1 \), it should reject \( H_0 \) with probability 1 under \( f_1 \), ensuring maximum power. 
Therefore, \( E_{f_1}(\phi(X)) = 1 \), which corresponds to option (B).