I am trying to find the UMVUE of $Exp(\frac{1}{\beta})$ using the fact that the distribution is complete.
I can do it by showing that $X$ is in the regular exponential class, but I also want to know how to do that using the definition of a complete distribution.
So, I am to prove that
$$E[u(X)]=0 \quad \text{iff} \quad u(X)=0$$
and this is what I have done so far.
$$E[u(X)] = \int_0^\infty u(x) \beta e^{-\beta x} dx = 0$$
so taking the derivative multiple times I was able to show that
$$E[X^ku(X)]=0$$
for $K=0,1,2,...$
Here, my notes say that
"$u(X)$ can be expanded based on $1, x, x^2, ...$" which is a polynomial congruent to $0$, thus all coefficients must be $0$ thus $u(X)=0$.
I am assuming that she wants me to expand $u(X)$ using Taylor's expansion, but what I get is
$$u(X) = u(0) + xu'(0)+ \frac{x^2}{2}u''(0)+...$$
which remotely looks like what I have shown but it is not exactly the same and am not comfortable moving forward in this direction.
Can someone help me out with this situation?
Thank you for your help.