You first find the suff stat, which is any multiple of .
Then you want to find an unbiased estimator of your 'parameter' that is based on this sum.
The thing you want to estimate is .
The expected value of S is n/p. You can try S^c and see what you get, but the first thing I would note is that
S is a sum of geo's, hence it's a negative binomial.
To use the Rao-Blackwell Theorem directly let , so T is unbiased for that probability.
The UMVUE will be .
See example 9.1 from http://www.stat.unc.edu/faculty/cji/lecture9.pdf for the messy next steps.
So you need
We know since X_1 is a geo.
And via is a NB where you are waiting for the success.
via is a NB where you are waiting for the success.