Assume $X_1,...,X_n$ is sample of $N(0,\sigma^2), \sigma > 0$, find
(1) complete and sufficient statistic of $\sigma^2$,
(2) UMVUE(uniformly minimum variance unbiased estimation) of $\sigma$ and $3\sigma^2$.
The problem is from my textbook.
Part (1) is solved, my answer is $T= \sum_{i=1}^n X_i^2$.
I aim to solve part (2) by applying Lehmann-Scheffe theoroem.
I need to find a function $g(T)$ of $T$ s.t. $Eg(T) = \sigma$.
And distribution of $T/\sigma^2$ is $\chi^2_n$. Hence I get $$ T \sim f(t) = \frac1{\sigma^n 2^{n/2} \Gamma(n/2)} t^{n/2-1}{\rm e}^{-\frac{t}{2\sigma^2}} I_{(t>0)} $$
Therefore $$ Eg(T) = \int_0^\infty \frac{g(t)}{\sigma^n 2^{n/2} \Gamma(n/2)} t^{n/2-1}{\rm e}^{-\frac{t}{2\sigma^2}} {\rm d}t = \sigma $$
I got stuck here to figure $g(t)$ out. Any hint is appreciated.
By the hint of comment,
$$ \begin{align} E\frac{T}{\sigma^2} &= n \\ E\frac{\sqrt{T}}{\sigma} &= \sqrt 2 \frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac n2)} \end{align} $$ Hence, $E\frac{3}{n}T = 3\sigma^2, E\sqrt2 \frac{\Gamma(\frac n2)}{\Gamma(\frac{n+1}{2})} T = \sigma $.
I get unbiased estimator of $3\sigma^2$ and $\sigma$, and they are functions of $T$.
By L-S theoroem, UMVUE of $3\sigma^2$ is $\frac3n T$, and UMVUE of $\sigma$ is $\sqrt2 \frac{\Gamma(\frac n2)}{\Gamma(\frac{n+1}{2})} T$.