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If $ X_1, ... ,X_n$ are IID binomial with parameters $ n$ and $p, $ find an unbiased estimator for

$$G(p)=P(X_1=1)=np(1-p)^{n-1}\, .$$

I need to find this estimator so I can apply Lehmann-Scheffé theorem. I think it can't be a "crazy" estimator. But I couldn't find anything. Can I have a hint?

$$\mathrm{E}[X_1(1-X_2/n)] = n p (1-p) \, .$$

Maybe it is a good way to follow.

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If $X_1,\dots,X_n$ are IID $\mathrm{Bin}(n,p)$, then for any bijection $\pi:\{1,\dots,n\}\to\{1,\dots,n\}$ we have $$ \mathrm{E}\!\left[X_{\pi(1)}\prod_{i=2}^n\left(1-\frac{X_{\pi(i)}}{n}\right)\right] = np(1-p)^{n-1} \, . $$ Just pick one of these $n!$ unbiased estimators.

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