# Best estimator for the variance of the empirical median

Let $$X_i$$ be $$n=2k+1$$ IID continuous random variables with distribution function $$F$$ and quantile function $$F^{-1}$$.

Let $$m$$ be their empirical median. It is well known that:

• If the $$X_i$$ were uniform on the $$[0,1]$$ interval, then $$m$$ would have a beta $$\beta(k,k)$$ distribution.

• The median of uniform variables is thus asymptotically Gaussian with variance $$var(m) = k/8 \approx n/4$$.

• If the $$X_i$$ are not uniform, then the median is instead a transformed version of the $$\beta(k,k)$$ using the quantile function:

$$m = F^{-1}( \beta_{k,k} )$$

• Using the delta method, the median is thus generally asymptotically Gaussian with variance $$var(m) = n/4 f(m_0)^2$$ where $$f(m_0)$$ is the density at the true median of the distribution $$F$$.

Thus, when we have complete knowledge of the underlying distribution, it is quite easy to characterize the distribution of $$m$$.

My question concerns the statistical problem of estimating $$var(m)$$ in the absence of such knowledge. Is there an accepted best method or at least papers trying to tackle this issue?