Question

Let \(X\) be a single observation from a distribution having a probability density function \(f_\theta\), \(\theta \in \Theta\). For testing \(H_0: \theta = 1\) against \(H_1: \theta \neq 1\), under which of the following options a uniformly most powerful test of level \(0.05\) exists?

• A. \[ f_\theta(x) = \begin{cases} \dfrac{1}{\theta} & \text{if } 0 < x < \theta \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (0, \infty) \]
• B. \[ f_\theta(x) = \begin{cases} e^{-(x-\theta)} & \text{if } x > \theta \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (-\infty, \infty) \]
• C. \[ f_\theta(x) = \begin{cases} \dfrac{1}{\theta} e^{-x/\theta} & \text{if } x > 0 \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (0, \infty) \]
• D. \[ f_\theta(x) = \begin{cases} \dfrac{1}{\theta} e^{-(x-1)/\theta} & \text{if } x > 1 \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (0, \infty) \]

Hint:

A UMP test for a two sided hypothesis can exist only when the parameter also fixes the boundary of the support of \(X\), because then one side of the alternative is detected for free.

Answer 1

The Correct Option is A, B

Step 1: General obstacle.
Two-sided tests usually have no UMP; best test for each direction differs.
Step 2: Special structure that fixes this.
Support boundary depending on \(\theta\) lets one direction be free.
Step 3: (A) test.
Support \((0,\theta)\), one-sided info free, UMP exists. TRUE.
Step 4: (B) test.
Support \((\theta,\infty)\), UMP exists. TRUE.
Step 5: (C),(D) test.
Support fixed regardless of \(\theta\), no free direction, no UMP. FALSE both.
\[ \boxed{\text{(A) and (B)}} \]