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.: $$E[\overline{\mu}] \neq \mu$$.

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

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