While doing some simulations, I realised that the sample quantile is a biased estimator of the true quantile. And, according to my simulations, a potentially very biased one.
I was surprised with that result since the empirical CDF isn't biased, but after some internet research, I discovered it was true.
I tried to figure out where that bias comes from, but working with sample quantiles is quite difficult. Does anyone have a demonstration of that bias (and, ideally, a quantification)?