The median of a finite set of data depends on the data in a continuous, but non-smooth way; this can be demonstrated by considering the median of the data set $\{ 0, 1, x \}$ as $x$ varies.
I wanted to know whether people have derived median-like statistics which depend smoothly on the data. By median-like, I mean roughly that
- they recover location parameters,
- they have some level of robustness associated to them, and ideally,
- they recover the median in some limit.
For example, it would be nice to have a parametrised family of functions $\{ \text{med}_\varepsilon \}_{\varepsilon > 0}$ which are each median-like, and such that as $\varepsilon \to 0$, one has that $\text{med}_\varepsilon ( \mathcal{D} ) \to \text{median} ( \mathcal{D} )$.
To clarify: I am particularly interested in knowing whether this exists in the literature already. If people have their own ideas for how to derive statistics with these properties, I'll gladly hear them out, but I would prefer to hear about tried-and-tested ideas.