Question

Let \(X_1,X_2,\ldots,X_n\ (n>1)\) be a random sample from the following probability density function
\[ f_{\beta}(x)=\begin{cases}\beta e^{-x}(1-e^{-x})^{\beta-1} & \text{if } x>0\\ 0 & \text{otherwise},\end{cases} \]
where \(\beta>0\) is an unknown parameter. For testing the following hypotheses,
\[ H_0:\beta=1 \quad \text{against} \quad H_1:\beta>1, \]
at level \(\alpha\in(0,1)\), which of the following statements is correct?

• A. The uniformly most powerful test does not exist
• B. For some constant \(a\), the critical region of the uniformly most powerful test will be of the form \(C=\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n x_i>a\}\)
• C. For some constant \(a\), the critical region of the uniformly most powerful test will be of the form \(C=\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n \ln(1-e^{-x_i})>a\}\)
• D. For some constant \(a\), the critical region of the uniformly most powerful test will be of the form \(C=\{(x_1,x_2,\ldots,x_n):\sum_{i=1}^n \ln(1-e^{-x_i})<a\}\)

Hint:

Substitute \(u=1-e^{-x}\) to see that \(f_{\beta}\) is a power (Beta) family in \(u\); this exponential family has MLR in \(\sum\ln(1-e^{-x_i})\), and Karlin-Rubin gives the UMP test directly.

Answer 1

The Correct Option is C

Step 1: Fix a single alternative value.
Pick $\beta_1>1$, compare $H_0:\beta=1$ vs simple alternative $\beta=\beta_1$. Neyman-Pearson rejects for large $L(\beta_1)/L(1)$.

Step 2: Compute this ratio.
\[ \frac{L(\beta_1)}{L(1)} = \beta_1^n \exp\left[(\beta_1-1)\sum_{i=1}^n \ln(1-e^{-x_i})\right]. \]

Step 3: Note this is increasing in $T$ for every $\beta_1>1$.
Same rejection rule $T>a$ is most powerful against every $\beta_1>1$.

Step 4: UMP exists.
Since the same rule works for the whole composite alternative, a UMP test exists, ruling out (A).

Step 5: Compare with wrong forms.
Statistic is $\sum\ln(1-e^{-x_i})$, not $\sum x_i$; test flags large $T$ not small $T$.

Final Answer:
\[ \boxed{C=\left\{(x_1,\ldots,x_n):\sum_{i=1}^n\ln(1-e^{-x_i})>a\right\}} \]