Solved – UMVUE for $theta$ parameter of geometric distribution

Let $X_1,X_2$ be a sample from $Geo(theta)$ ($0<theta<1$).

I need to find the UMVUE for $theta$.

I started with establishing that $Y=X_1+X_2$ is a sufficient complete statistic for $theta$ and $Ysim NB(2,theta)$.

From the Lehmann–Scheffé theorem, this means that Y is UMVUE for $E(Y)=frac{2}{theta}$.

But that is not what I need. What should be my next step?

I also know that the MLE of $theta$ is $frac{2}{Y}$. Since it is a function of $Y$ alone, it should be UMVUE for $E(frac{2}{Y})$ but how do I compute this to check if it is $theta$?

As a different approach, consider the estimator,

$$ T(X_1, X_2) = begin{cases}1 & X_1 = 1\ 0 & otherwise end{cases} $$

This is unbiased for $theta$ (why?). Then, by Lehmann-Scheffe, condition on your complete sufficient statistic, $E[T(X_1,X_2) | X_1 + X_2]$ to get the UMVUE. Finding the exact closed form of $E[T(X_1,X_2) | X_1 + X_2]$ will take a tiny bit of work though.

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