Best estimator for median

Question

I'm wondering how we can estimate the median from sample data? I know when the distribution is normal, the mean is an unbiased estimator for the median. Is this also true when the distribution is only symmetric? What if the distribution is not symmetric at all.

Answer 1

Yes, I believe that a symmetric distribution with finite mean has median equal to its mean. The mode is also the same if the distribution is unimodal.

On the other hand, you can look at the sample median (rather than the sample mean) as an estimator for the median. The sampling distribution of the sample median looks asymptotically like: $$ \hat{m}_n\sim\mathcal{N}\left( m, [4n f(m)^2]^{-1} \right) $$ where $m$ is the median, $f$ is the density, and $\hat{m}_n$ is the sample median with $n$ samples. There is a "central limit theorem" for the sample median as well.

This suggests, I think, that, with sufficient samples, using the sample median can be a good estimator for the median even in the asymmetric case. Since $m=\mu$ in the symmetric case, you can use the sample mean there.

See also: [1], [2], [3], [4].