I have a an iid sample from a geometric (p) distribution. I found that the complete sufficient statistic is using "full exponential family with open set."
I am trying to find the UMVUE of .
I am arguing that the expectation of 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!