Finding a UMVUE for variance of normal distribution

Clarkson

New Member
Let Let $$X_1,X_2,...,X_n$$ be a random sample from a normal distribution with mean $$\mu$$and variance $$\sigma^2$$.
I showed that $$(\bar X,S^2)$$ is jointly sufficient for estimating ( $$\mu$$, $$\sigma^2$$) where $$\bar X$$is the sample mean and $$S^2$$ is the sample variance.

Then assuming that $$(\bar X,S^2)$$ is also complete I have to show that $$\sqrt{ n-1\over 2}{\Gamma ({ n-1\over 2})\over\Gamma (\frac n2)} S$$
is a Uniformly Minimum Variance Unbiased Estimator for $$\sigma$$.

I think I have to use Lehman Scheffe theorem as $$(\bar X,S^2)$$is jointly sufficient and complete for $$\sigma$$.
But how can I find a function which is unbiased for $$\sigma$$ that contains both $$(\bar X,S^2)$$.

I don't understand how to work when there's a joint sufficiency and completeness.

BGM

TS Contributor
But how can I find a function which is unbiased for $$\sigma$$ that contains both $$(\bar X,S^2)$$.
The proposed estimator in the question already is a function of $$(\bar X,S^2)$$