A Curious Result on Breaking Ties among Sample Medians

07/10/2018

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by   Peter M. Aronow, et al.

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It is well known that any sample median value (not necessarily unique) minimizes the empirical L^1 loss. Interestingly, we show that the minimizer of the L^1+ϵ loss exhibits a singular phenomenon that provides a unique definition for the sample median as ϵ→ 0. This definition is the unique point among all candidate median values that balances the logarithmic moment of the empirical distribution. The result generalizes directly to breaking ties among sample quantiles.