# Expected value of the reciprocal of sample standard deviation

Hopefully I am posting this in the right place, I've never used StackExchange before.

The question I am answering is:

Assume that X1, . . . , Xn are iid from $$N(µ, σ^2)$$ with unknown µ and $$σ^2$$.

Find the best unbiased estimator of θ = µ/σ.

So, I know that $$(\bar{X}, S^2)$$ is a sufficient and complete statistic for $$(µ, σ^2)$$, and I know that $$\bar{X}$$ and $$S^2$$ are independent, so I want to find $$E[\frac{\bar{X}}{S}]=E[\bar{X}]E[\frac{1}{S}]$$.

Ideally, this will lead to an answer of $$\frac{\mu}{\sigma} \times$$ (some constant), then I can multiply $$\frac{\bar{X}}{S}$$ by the reciprocal of that constant to get the best unbiased estimator.

I am struggling to figure out how to do $$E[\frac{1}{S}]$$. I know that $$\frac{S^2(n-1)}{\sigma^2} \sim \chi^2_{n-1}$$, so I believe I need to use this, but I am stuck at that point. I have the answer, as this is just a practice question, but the answer doesn't explain how $$E[\frac{1}{S}]$$ was calculated.

Thank you

Note that $$S^2 \stackrel{d}{=} \frac{\sigma^2}{(n-1)}W$$ where $$W\sim\text{Gamma}(n-1/2, 2)$$ under the shape, scale parametrization. In particular $$EW^k=\frac{\Gamma((n-1)/2+k)}{\Gamma((n-1)/2)}2^k.$$ It follows that $$ES^{-1} =\frac{\sqrt{n-1}}{\sigma}EW^{-1/2} =\frac{\sqrt{n-1}}{\sigma\sqrt{2}}\frac{\Gamma((n-2)/2)}{\Gamma((n-1)/2)}\quad (n\geq 1).$$