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Suppose that $ (X_1,X_2, \ldots, X_n) $ be a random sample from Bernoulli (p). Find the UMVUE of

$$\tau(p)= (1-p)+ e^{2}p $$

My approach:

I know $T(X) = \sum_{i=1}^n X_i$ is a complete sufficient statistic for the Bernoulli distribution. And by Lehmann_Scheffe theorem,

$$\phi(T)= \mathbb{E}(W(X)\mid T(X)) $$

where W(X) is an unbiased estimator. Can one define $W(X)= (1-X_1)+ e^{2}X_1$?

How do I continue from here?

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  • $\begingroup$ What is your $W$? You have stated the theory without showing any work. $\endgroup$ – StubbornAtom Apr 7 at 6:37
  • $\begingroup$ I have re-edited the post to include the definition of W. I have no idea how to go about it. That's why I am asking for help. $\endgroup$ – Lady Apr 7 at 14:17
  • $\begingroup$ If $e^2$ is just a constant, $E\left[W\mid T\right]=E\left[(e^2-1)X_1+1\mid T\right]=(e^2-1)E\left[X_1\mid T\right]+1$. Now you have to deduce the distribution of $X_1\mid T$. $\endgroup$ – StubbornAtom Apr 7 at 15:08

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