Let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=\theta(1-\theta)^x,x=0,1,\cdots; 0 < \theta <1$ is unknown. Find the UMVUE of $\frac{1-\theta}{\theta}$.
My work:
I know that I should apply the Lehmann-Scheffe Theorem here, so I will need an unbiased estimator of $\frac{1-\theta}{\theta}$ and a complete sufficient statistic for $\theta$. I know that, since this distribution is an exponential family, the complete sufficient statistic is $\sum^n_{i=1}x_i$. However, I am having problems finding an unbiased estimator of $\frac{1-\theta}{\theta}$. Is there a systematic approach that I can take to finding an unbiased estimator?