Performance of the empirical median for location estimation in heteroscedastic settings

We investigate the performance of the empirical median for location estimation in heteroscedastic settings. Specifically, we consider independent symmetric real-valued random variables that share a common but unknown location parameter while having different and unknown scale parameters. Estimation under heteroscedasticity arises naturally in many practical situations and has recently attracted considerable attention. In this work, we analyze the empirical median as an estimator of the common location parameter and derive matching non-asymptotic upper and lower bounds on its estimation error. These results fully characterize the behavior of the empirical median in heteroscedastic settings, clarifying both its robustness and its intrinsic limitations and offering a precise understanding of its performance in modern settings where data quality may vary across sources.