Discussion
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be i.i.d. Bernoulli \((p)\) random variables and \(S = \sum_{i=1}^{n} X_{i}\). Then unbiased estimator of \(p^{3}\) is :
Answer: B
The unbiased estimator is \(\dfrac{S(S - 1)(S - 2)}{n(n - 1)(n - 2)}\).
\(S\) is binomial \((n, p)\), and its factorial moments are \(E[S(S - 1)(S - 2)] = n(n - 1)(n - 2)p^{3}\). Dividing by the constant gives an estimator with expectation \(p^{3}\). It is also the probability that three specified trials all succeed, \(X_{1}X_{2}X_{3}\), averaged over all triples, so it is the UMVUE by the Lehmann-Scheffe theorem.
\((S/n)^{3}\) is biased: \(E(S^{3})/n^{3}\) contains lower-order terms in \(p\) besides \(p^{3}\). The shifted form \((S - 1)(S - 2)(S - 3)/[(n - 1)(n - 2)(n - 3)]\) has the wrong factorial moment. Since an unbiased estimator exists, 'None of the above' is wrong.
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