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suppose $X_1,X_2,\ldots,X_n,X_{n+1}$ be a random sample of distribution $N(\theta,1)$. if $\gamma(\theta)=P_{\theta}(\displaystyle\sum_{i=1}^{n} X_i>X_{n+1})$ , how can i find UMVU estimator of parameter $\gamma(\theta)$

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What did you try? Where are you stuck? –  Did May 3 '12 at 15:02
The way these things generally go is write down any unbiased estimator & condition on a complete sufficient statistic. In this case $1_{(X_1 + ... + X_n > X_{n+1}0}$ is unbiassed and u would be looking for $\mathbb P (X_1 + ... + X_n > X_{n+1} \vert X_1 + ... + X_n + X_{n+1})$. As conditioning on the sum just converts the $X_i$ to normals with different distribution, it is do-able. –  mike May 3 '12 at 17:11

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