Determining the MVUE of $\theta$ when $f(x;\theta) = \theta^x (1-\theta)$

Question

The Statement of the Problem:

Let $X_1, X_2, ... , X_n$ be a random sample from

$$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$

(a) Find the ML estimator of $\theta$.

(b) Show that $T = \sum_{i=1}^n X_i$ is a sufficient statistic for $\theta$.

(c) Determine the MVUE of $\theta$.

Where I Am:

I've taken care of parts (a) and (b) but am a bit stuck on part (c). In order to find a MVUE, I have to find an unbiased estimator, which I can't seem to figure out. My MLE, which is the following:

$$ \hat\theta_{\text{mle}} = \frac{\sum_{i=1}^n X_i}{n + \sum_{i=1}^n X_i} $$

seems to be biased (although, honestly, I got stuck trying to find its expectation). So, I've tried making an unbiased estimator from scratch... with no luck. Any tips here would be helpful. Thanks.

Answer 1

Use Lehmann-Scheffe Theorem (or Rao-Blackwellization). You have $T(\bf{X}$$)=\sum X_i$ to be complete-sufficient. Now find a simple unbiased estimator $\delta_0=I(X_1>0)$. Yes with just $X_1$ only! Then find $E[\delta_o|T(\bf{X})]$. I think you can figure it out from here. Let me know if you've further doubt/query.