# Prove an estimator is biased

I want to prove that an estimator in biased. For a big n, like $$10^{13}$$ we define a distribution of values $$x_i \in (0,1]$$ for $$i \in [n]$$. We want to estimate with some error $$\epsilon$$ the value of:

$$\mu = \sum_{i}^n \log x_i$$

The distribution of $$x_i$$ the values of $$x_i$$ is not uniform on the interval: there are some (like $$10^2$$) values that are very close to one, and the rest are almost zero.

We are given $$m$$ samples of $$x_i$$ with uniform distribution, that is: $$p(x_i)=p(x_j)$$ for $$j\neq i$$.

Then, I compute $$\overline{\mu}=n \times \frac{1}{m}\sum_{i=1}^m \log x_i$$.

I want to show that $$\overline{\mu}$$ does is not the right estimator for $$\mu$$, and I think it is biased, i.e.: $$E[\overline{\mu}] \neq \mu$$.

• why? what's wrong with negative numbers? – asdf Feb 17 at 10:09
• Because of Jensen's inequality, your last equality is not valid. – Bertrand Feb 17 at 11:07
• Why do you think that this estimator is biased? It seems almost immediate that it's unbiased. – joriki Feb 17 at 13:02