Discussion
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample observed from exponential distribution with location parameter \(\theta\). For testing \(H_{0} : \theta = \theta_{0}\) against \(H_{1} : \theta \gt \theta_{0}\), the UMP test rejects \(H_{0}\), if :
where C is to be chosen suitably.
Answer: B
The UMP test rejects \(H_{0}\) if \(X_{(1)} \gt C\).
The density is \(f(x; \theta) = e^{-(x - \theta)}\) for \(x \gt \theta\), so the likelihood is
\[ L(\theta) = e^{-\sum x_{i} + n\theta} I(x_{(1)} \gt \theta) \]
It depends on \(\theta\) only through the smallest observation \(X_{(1)}\), which is sufficient. For \(\theta_{1} \gt \theta_{0}\), the ratio \(L(\theta_{1})/L(\theta_{0})\) is 0 when \(x_{(1)} \le \theta_{1}\) and the constant \(e^{n(\theta_{1} - \theta_{0})}\) otherwise, a non-decreasing function of \(x_{(1)}\). So the family has monotone likelihood ratio in \(X_{(1)}\), and the UMP test for the one-sided alternative rejects for large \(X_{(1)}\). C comes from \(n(X_{(1)} - \theta_{0})\) being exponential with mean 1 under \(H_{0}\).
\(\bar{X}\) and \(X_{(n)}\) are not sufficient for the location, and \(\sum(X_{i} - X_{(1)})\) is ancillary: its distribution does not depend on \(\theta\) at all, so it cannot detect a shift.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development