If a symmetric distribution is ε-contaminated and the contaminants have finite first moments, the median may cease to be the most robust estimator of location.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am29-1-1, author = {Ryszard Zieli\'nski}, title = {The most stable estimator of location under integrable contaminants}, journal = {Applicationes Mathematicae}, volume = {29}, year = {2002}, pages = {1-6}, zbl = {1053.62040}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am29-1-1} }
Ryszard Zieliński. The most stable estimator of location under integrable contaminants. Applicationes Mathematicae, Tome 29 (2002) pp. 1-6. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am29-1-1/