Discussion
Let \(S = S(X_{1}, X_{2}, ......, X_{n})\) be an unbiased estimator of the parametric functional \(\theta(F)\) based on a random sample \(X_{1}, X_{2}, ......, X_{n}\) from F. Suppose U is the U-statistics corresponding to S. Then, which of the following statements is true ?
Answer: C
The true statement is \(V_{F}(U) \le V_{F}(S)\).
The U-statistic corresponding to S is its average over all permutations of the sample, which is the conditional expectation of S given the order statistics: \(U = E[S \mid X_{(1)}, \ldots, X_{(n)}]\). The order statistics are sufficient for an i.i.d. sample from an unrestricted F.
So U has the same expectation as S, \(\theta(F)\), which rules out A and B, and by the Rao-Blackwell argument (conditional variance formula)
\[ V(S) = V(U) + E[V(S \mid \text{order stats})] \ge V(U) \]
So the U-statistic never has larger variance, and D is wrong.
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