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?



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