# Finding best unbiased estimator of ratio of mean to std.dev ($\frac{\mu}{\sigma}$) from normal population with unknown parameters.

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.

• Start from $\overline X/S$. Commented Mar 20, 2020 at 6:07
• I'm getting something very complicated. $E(\frac{\bar{X}}{S}))=\mu E(S^{-1})$ which I calculate to be $\frac{\mu}{\sigma}\sqrt{2(n-1)}\frac{\Gamma(n/2)}{\Gamma(n-1/2)}$ Commented Mar 20, 2020 at 7:01
• @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/…
– NCh
Commented Mar 20, 2020 at 7:24
• Note that the answer must be smth like you wrote, with gamma-functions. So almost sure you are right.
– NCh
Commented Mar 20, 2020 at 7:25
• @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. Commented Mar 20, 2020 at 7:57

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:
$$E(1/S)=E(\frac{\sqrt{n-1}}{\sigma}Y^{-1/2})=\frac{\sqrt{n-1}}{\sigma}E(Y^{-1/2})$$
$$E(Y^{-1/2})=(\Gamma(\frac{n-1}{2})2^{\frac{n-1}{2}})^{-1}\int_{R^+}s^{1-\frac{n}{2}}e^{-s/2}ds=(\Gamma(\frac{n-1}{2})2^{\frac{n-1}{2}})^{-1}\Gamma(\frac{n}{2})2^{n/2}$$
=$$\sqrt2\frac{\Gamma({\frac{n}{2}})}{\Gamma({\frac{n-1}{2}})}$$
• The integral $\int_{R^+}s^{1-\frac{n}{2}}e^{-s/2}\,ds$ should be $\int_{R^+}s^{\color\red{\frac{n}{2}-1-1}}e^{-s/2}\,ds$. Commented Mar 20, 2020 at 12:07