Discussion
Suppose that \(X_{1}\) and \(X_{2}\) are independent random variables having Bernoulli distribution with mean \(p\) and \(2p\) respectively, where \(0 \lt p \lt 1/2\). The following are two statements :
(I) \((X_{1}, X_{2})\) is minimal sufficient for \(p\)
(II) \(2X_{1} + 5X_{2}\) is minimal sufficient for \(p\)
Which of the following is true ?
Answer: C
Both (I) and (II) are true.
The joint pmf is \(p^{x_{1}}(1 - p)^{1 - x_{1}}(2p)^{x_{2}}(1 - 2p)^{1 - x_{2}}\). The four sample points give likelihoods \((1 - p)(1 - 2p)\), \(p(1 - 2p)\), \(2p(1 - p)\) and \(2p^{2}\). No two of these have a ratio free of \(p\); for instance \(p(1 - 2p)/[2p(1 - p)] = (1 - 2p)/(2(1 - p))\) depends on \(p\). So no two points can be merged, and the full sample \((X_{1}, X_{2})\) is minimal sufficient.
The statistic \(2X_{1} + 5X_{2}\) takes the values 0, 2, 5 and 7 on the four points, one value per point, so it is a one-to-one function of \((X_{1}, X_{2})\) and is minimal sufficient too.
The trap is to assume a minimal sufficient statistic must reduce the data. Here the sum \(X_{1} + X_{2}\) is not even sufficient, because it merges (1, 0) and (0, 1), whose likelihoods have different shapes. So the options denying (I) or (II) are wrong.
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