$\overline{x}$ is an unbiased estimator of the exact average.
Now, we let's imagine that we want to estimate some function of the average $f(\langle x \rangle)\equiv f(X)$.
My first guess was $\overline{f(x)}=\frac{1}{N}\sum f(x_i)$. However, when you analyse the bias one obtains $\langle\overline{f(x)}\rangle-f(X)=\int P(x)(f(x)-f(X))dx=f'(X)\int P(x)(x-X)dx+\frac{1}{2}f''(X)\int P(x)(x-X)^2dx+...=\frac{1}{2}f''(X)(\langle x^2 \rangle - \langle x \rangle^2)=\frac{1}{2}f''(X)\sigma^2$
We get that the estimator is of order unity, so if you increase $N$, the bias remains the same.
I want a best estimator and its proof. I know it's $f(\overline{x})$, however I don't follow the proof (it seemes so straight forward)
The document that I'm following is http://young.physics.ucsc.edu/jackboot.pdf