The empirical median for estimating the common mean of heteroscedastic random variables

We study the problem of mean estimation in the heteroscedastic setting. In particular, we consider symmetric random variables having the same location parameter and different and unknown scale parameters. Our goal is then to estimate their unknown common location parameter. It is an elementary topic but yet a not very well-studied one since we always make the assumption that the random variables are independent and identically distributed. In this paper, we study the median estimator and we establish upper and lower bounds on its estimation error that are of the same order and that generalize and improve recent results of Devroye et al. and Xia.