Question

Rao-Blackwell theorem enables us to obtain minimum variance unbiased estimator through :

• A. An unbiased statistic
• B. A sufficient statistic
• C. A complete statistic
• D. An efficient statistic

Hint:

Rao-Blackwellization means conditioning an unbiased estimator on a sufficient statistic to shrink its variance.

Answer 1

The Correct Option is B

Step 1: Suppose \(\phi\) is any unbiased estimator of a parametric function \(\tau(\theta)\), and \(T\) is a sufficient statistic for \(\theta\).

Step 2: The Rao-Blackwell theorem defines a new estimator \(\phi^{*} = E[\phi \mid T]\), the conditional expectation of \(\phi\) given the sufficient statistic \(T\).

Step 3: Because \(T\) is sufficient, \(\phi^{*}\) does not depend on \(\theta\) and is a genuine statistic; it remains unbiased for \(\tau(\theta)\) and satisfies \(\text{Var}(\phi^{*}) \le \text{Var}(\phi)\).

Step 4: So conditioning any unbiased estimator on a sufficient statistic never increases variance, and this construction is exactly the mechanism the Rao-Blackwell theorem uses to move toward a minimum variance unbiased estimator.

Final Answer: (B) A sufficient statistic.