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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

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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 $$ ES^{-1} =\frac{\sqrt{n-1}}{\sigma}EW^{-1/2} =\frac{\sqrt{n-1}}{\sigma\sqrt{2}}\frac{\Gamma((n-2)/2)}{\Gamma((n-1)/2)}\quad (n\geq 1). $$

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  • $\begingroup$ This helps me a lot, I did not realize the connection I could make between the Chi-squared and Gamma distributions. Thank you so much!!! (I tried to upvote this, but it says I need at least 15 reputation points until I can do that) $\endgroup$
    – jskarmeas
    Jun 13, 2023 at 15:27

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