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We were asked to find the UMVUE for $\lambda$ using the Lehmann-Scheffe Theorem, given a random sample from $f_X(x;\lambda)=\frac{1}{\lambda}x^{\frac{1}{\lambda}-1}, 0<x<1, \lambda > 0$.

I know that in order to use the theorem, I need to find an unbiased estimator, and a complete sufficient statistic for $\lambda$. I already found a c.s.s. but I am quite lost on how to find the unbiased estimator.

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Let $X_1,\dots,X_n$ be a random sample from a $\text{Beta}(1/\lambda,1)$ distribution. Hence, $X_i$ has the desired density: $f(x_i\mid\lambda)=(1/\lambda)\,x_i^{(1/\lambda)-1}$, for $0<x<1$, and $\lambda>0$. It's not difficult to show that $\mathbb{E}[\log X_i]=\psi(1/\lambda)-\psi(1+1/\lambda)=\lambda$, in which $\psi$ is the digamma function. It follows that $\hat{\lambda}=(1/n)\sum_{i=1}^n \log X_i$ is an unbiased estimator of $\lambda$. By the way, this is the maximum likelihood estimator of $\lambda$, as you can easily verify.

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