# Confidence on the median for a weighted empirical distribution

I have observation data $$x_1, \dots, x_n$$ with associated positive weights $$w_1, \dots, w_n$$ such that $$\sum_i w_i = 1$$, representing a "weighted" empirical distribution with CDF $$\hat{F}(x) = \sum_{i=1}^n w_i I(x_i \le x)$$ where $$I$$ is the indicator function. This generalizes the usual empirical distribution, which is the special case $$w_i = 1/n$$. Is there a closed-form expression for a confidence interval for the median in this case?

I'm aware that this question has been answered before for the case $$w_i = 1/n$$, for which the confidence interval can be obtain from the binomial distribution, but it's not obvious to me how to treat general case. I realize that bootstrap estimators are possible, but I'm interested in deriving an analytical expression, if possible.

• You need to specify the distribution of the $x_i$'s to hope for a solution. – Xi'an Feb 5 at 17:34
• @Xi'an Possibly, but can you show why? Given that there is a distribution-free solution for the case $w_i = 1/n$, why isn't there a solution for arbitrary $w_i$ ? – Roland Feb 5 at 17:39
• Because the weights need be connected with the distribution of the $x_i$'s. – Xi'an Feb 5 at 21:08