Question

Match List - I with List - II. 

List - I	List - II
A. Minimum variance bound estimator	I. Cramer-Rao Inequality
B. Sufficient statistics	II. Factorisation Theorem
C. UMVU estimator	III. Rao-Blackwell Theorem
D. Unique UMVUE	IV. Complete sufficient statistics

• A. A-IV, B-I, C-II, D-III
• B. A-IV, B-III, C-II, D-I
• C. A-II, B-III, C-I, D-IV
• D. A-II, B-III, C-IV, D-I

Hint:

Keywords: Factorization $\leftrightarrow$ Sufficiency; Cramer-Rao $\leftrightarrow$ Variance Bound; Rao-Blackwell $\leftrightarrow$ UMVUE improvement.

Answer 1

The Correct Option is C

We match the statistical concepts in List I with their corresponding theorems or criteria in List II.

Step 1: \color{redMatch A (Minimum variance bound)
The Cramer-Rao Inequality provides a lower bound for the variance of unbiased estimators. An estimator that achieves this bound is called a Minimum Variance Bound (MVB) estimator.
Match: A-II.

Step 2: \color{redMatch B (Sufficient statistics)
The Neyman-Fisher Factorisation Theorem is the standard criterion used to identify whether a statistic is sufficient for a parameter.
Match: B-III.

Step 3: \color{redMatch C (UMVU estimator)
Complete sufficient statistics are central to the Lehmann-Scheffé theorem, which is used to find Uniformly Minimum Variance Unbiased Estimators (UMVUE).
Match: C-I.

Step 4: \color{redMatch D (Unique UMVUE)
The Rao-Blackwell Theorem shows how to improve an estimator by conditioning it on a sufficient statistic, leading toward the UMVUE. Combined with completeness, it ensures uniqueness.
Match: D-IV.

Step 5: \color{redFinal Matching
A-II, B-III, C-I, D-IV.
This corresponds to Option (3).