I am prepping for an exam and this is one of the questions the professor handed to help us prepare.
$$X\sim NB(k,p)$$ where $k$ is the number of successes. Find the UMVUE of $g(p)=p^k$ if it exists.
Note $X$ is a minimally sufficient complete statistics so must find:
$$\sum_{t=k}^{\infty}T(t){t-1\choose k-1}p^k(1-p)^{t-k}=p^k$$
This implies: $$\sum_{t=k}^{\infty}T(t){t-1\choose k-1}(1-p)^{t}=(1-p)^k$$
Note that the infinite sum on the right cannot replicate the polynomial on the right since their powers will differ. Thus the only UMVUE is $T(X)=I(X=k)$ to guarantee equal powers on both sides. Is my logic correct? This will only work if $p\in(0,1)$ not at the end points right?