Discussion
If \(T_{1}\) is an unbiased estimator of \(\theta\) and T is a minimal sufficient statistic for \(\theta\), then which of the following statements is NOT true ? Suppose \(T^{*} = E[T_{1} \mid T]\).
Answer: C
The statement that is NOT true is \(\mathrm{Var}(T^{*}) \le \mathrm{Var}(U)\) for every unbiased estimator U of \(\theta\) (without completeness).
Rao-Blackwellisation, \(T^{*} = E[T_{1} \mid T]\), always improves on the estimator it starts from: \(T^{*}\) is unbiased and \(\mathrm{Var}(T^{*}) \le \mathrm{Var}(T_{1})\). That makes options B and D true.
Beating every unbiased estimator needs more. By the Lehmann-Scheffe theorem, if T is complete as well as sufficient, \(T^{*}\) is the unique UMVUE, which is option A. Minimal sufficiency alone does not give completeness, and then different unbiased starting points can lead to different conditional expectations, so option C can fail.
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