calculating UMVUE of parameter $(1-\sigma^2)^-\frac{n}{2}$.

Question

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^-\frac{n}{2}$. I know $T=\sum_{i=1}^n X_i^2$ is Sufficient and complete statistics for $\sigma^2$.therefore $E(g(T))= (1-\sigma^2)^-\frac{n}{2}$

Answer 1

Since $E[X_i^2]=\mu^2 + \sigma^2=\sigma^2$, $\frac{T}{n}$ is unbiased for $\sigma^2$ and is also complete, as you said. So, since expectation is a linear operator, $\frac{-n}{2}(1-\frac{T}{n})$ is unbiased for $\frac{-n}{2}(1-\sigma^2)$. Unbiased estimators which are functions of complete and biased statistics are UMVUE.