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: Transform to recognize the family.
Let \(U=1-e^{-X}\). Substituting gives the density of \(U\): \[ f_U(u)=\beta u^{\beta-1},\quad 0<u<1. \] Standard power (Beta\((\beta,1)\)) distribution.

Step 2: Write the joint density as exponential family.
\[ f(x_1,\ldots,x_n;\beta)=\beta^n\left(\prod_{i=1}^n e^{-x_i}\right)\exp\left[(\beta-1)\sum_{i=1}^n\ln(1-e^{-x_i})\right]. \] Natural parameter \(\eta(\beta)=\beta-1\), sufficient statistic \(T=\sum_{i=1}^n\ln(1-e^{-x_i})\).

Step 3: Apply MLR.
For \(\beta_2>\beta_1\), the likelihood ratio is increasing in \(T\), so this family has MLR in \(T\).

Step 4: Karlin-Rubin.
UMP level-\(\alpha\) test exists, rejects when \(T>a\). Option (A) is FALSE.

Step 5: Rule out (B).
Sufficient statistic is \(\sum\ln(1-e^{-x_i})\), not \(\sum x_i\).

Step 6: Rule out (D).
Larger \(\beta\) makes \(T\) larger, so reject for large \(T\), not small.

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