Math Help - UMVUE of geometric sample

1. UMVUE of geometric sample

Hi,

I have a an iid sample from a geometric (p) distribution. I found that the complete sufficient statistic is $\sum_{i=1}^n x_i$ using "full exponential family with open set."

I am trying to find the UMVUE of $\tau (p) = E_p {X_1}$.

I am arguing that the expectation of $X_1$ equals the expectation of my complete sufficient statistic and thus I use it as my estimator and it is unbiased. Furthermore, because the unbiased estimator is also complete, I am guaranteed that it is UMVUE.

Is that a tight argument or do I actually need to go through the steps of finding the conditional expectation? (I did find the related post of http://www.mathhelpforum.com/math-help/advanced-probability-statistics/86748-need-help-statistics-problem.html)

I'd appreciate any feedback! Thank you!

2. parameters to be estimated

Hi,

Am I on the right track by noting that the parameter that I am trying to estimate is 1/p, while the discussion in the link is trying to estimate p?

Thank you!