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suppose $X_1,X_2,\ldots,X_n$ is a random sample of $Beta(\theta+1,1)$. how can find UMVUE of parameter $\displaystyle 3^{-\theta}$

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closed as off-topic by Did, Dario, Najib Idrissi, Davide Giraudo, Claude Leibovici Feb 22 '15 at 10:33

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Let $a$ in $(0,1)$, then $T=\mathbf 1_{X_1\lt a}$ is such that $\mathbb E_\theta(T)=a\cdot a^\theta$. Since $U=X_1\cdots X_n$ is a complete sufficient statistic, the unique UMVUE for $a^\theta$ is $S=a^{-1}\mathbb E_\theta(T\mid U)=a^{-1}\mathbb P_\theta(X_1\lt a\mid U)$.

Can you compute this conditional probability?

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