Asymptotic expansions for median estimate of a parameter

We obtain asymptotic expansions for the distribution of a median (of empirical distribution) estimate of a parameter in additive noise with symmetric density. For Laplacian (i.e., two-sided exponential) density this estimate coincides with the maximum likelihood estimate. As a corollary we obtain asymptotic expansions for moments of these estimates. Numerical comparisons with exact data show that the use of asymptotic expansions significantly increases the accuracy of statistical inferences even for relatively small sample sizes.