Let $X_1....X_n$ be a random sample from a $N(\mu,\sigma^2)$ population with unknown mean and variance. Find the BUE (best unbiased estimator) for $\frac{\mu}{\sigma}$.
I think that $T(X)=(\bar{X}, S^2)$ is complete and sufficient. They also happen to be independent which is nice. The next step is usually to find $g(T)$ s.t. $E(g(T))=(\mu/\sigma)$ but I'm stuck here. I cannot even think of an unbiased estimator of $\mu/\sigma$ that does not depend on T.
Hints and/or solutions both fine.
My attempt. We seek $E(1/S)$. We know $S^2$ $\overset{\mathcal{D}}{=} Y\frac{\sigma^2}{n-1}$ where $Y \sim \chi^2(n-1)$ The calculation goes:
$$E(1/S)=E(\frac{\sqrt{n-1}}{\sigma}Y^{-1/2})=\frac{\sqrt{n-1}}{\sigma}E(Y^{-1/2})$$
$$E(Y^{-1/2})=(\Gamma(\frac{n-1}{2})2^{\frac{n-1}{2}})^{-1}\int_{R^+}s^{1-\frac{n}{2}}e^{-s/2}ds=(\Gamma(\frac{n-1}{2})2^{\frac{n-1}{2}})^{-1}\Gamma(\frac{n}{2})2^{n/2}$$
=$$\sqrt2\frac{\Gamma({\frac{n}{2}})}{\Gamma({\frac{n-1}{2}})}$$
Is there a mistake?