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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.

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  • $\begingroup$ Start from $\overline X/S$. $\endgroup$ – StubbornAtom Mar 20 at 6:07
  • $\begingroup$ I'm getting something very complicated. $E(\frac{\bar{X}}{S}))=\mu E(S^{-1})$ which I calculate to be $\frac{\mu}{\sigma}\sqrt{2(n-1)}\frac{\Gamma(n/2)}{\Gamma(n-1/2)}$ $\endgroup$ – Muselive Mar 20 at 7:01
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    $\begingroup$ @Muselive It looks very (very!) believable, but too lazy to check. Here are the answers in a similar task - but with squares it's easier: math.stackexchange.com/questions/2684640/… $\endgroup$ – NCh Mar 20 at 7:24
  • $\begingroup$ Note that the answer must be smth like you wrote, with gamma-functions. So almost sure you are right. $\endgroup$ – NCh Mar 20 at 7:25
  • $\begingroup$ @Muselive It is supposed to be that kind of an answer. I think it should be $E\left[c_n\frac{\overline X}{S}\right]=\frac{\mu}{\sigma}$ where $c_n=\frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n-2}{2})}\sqrt{\frac{2}{n-1}}$. Previously asked: math.stackexchange.com/q/2063357/321264. $\endgroup$ – StubbornAtom Mar 20 at 7:57
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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?

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    $\begingroup$ The integral $\int_{R^+}s^{1-\frac{n}{2}}e^{-s/2}\,ds$ should be $\int_{R^+}s^{\color\red{\frac{n}{2}-1-1}}e^{-s/2}\,ds$. $\endgroup$ – StubbornAtom Mar 20 at 12:07
  • $\begingroup$ True, I messed up gamma density. $\endgroup$ – Muselive Mar 21 at 19:32

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