Large sample and robust properties of $$tilde L^2 $$-median

As a competitive depth, $ tilde L^2 $ tilde L^2 -depth is modified from L ²-depth. Its induced median is called $ tilde L^2 $ tilde L^2 -median. Basic properties of the median and its sample version are provided. Especially, the strong consistency of sample median is gained under weaker condition. Robustness of the median and its sample version is discussed. Besides ease of computation, it is shown that $ tilde L^2 $ tilde L^2 -median has both good large-sample and robust properties. Simulation studies are also given to compare the breakdown point of $ tilde L^2 $ tilde L^2 -median with that of other depth-induced medians.