While doing some statistics exercises I found a question that I don't know how to solve. The question is as follows:
Let $X_{1},...X_{n} \stackrel{iid}{\sim} N(0,\sigma^2)$, where $\sigma^2$ $\in R^{+}$ is unknown and density function:
$f_{X_{i}} (x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^-\frac{x^2}{2 \sigma^2}$, $x \in R$
a) find an unbiased estimator for $\sigma^2$ based only on the first observation $X_{1}$ and determine its variance.
My thoughts: I know that the first observation (i.e. the smallest) is computed as $X_{1} = n (1-F(x))^{n-1} f(x)$, but I need to compute the integral of the above density function, but is there an easier way? Can you give me some hints?
Consider the estimator $X_1^2$. It is unbiased since $$\mathsf{E}X_1^2=\sigma^2$$ Its variance is also $$\mathsf{Var}(X_1^2)=\mathsf{E}X_1^4-\mathsf{E}^2X_1^2=3\sigma^4-\sigma^4=2\sigma^4$$