*I previously asked this on [here][1] on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO.* 

**Background**

notation: RV= random variable, $\mu=$ mean $m=$ median

Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.

If $f(x)$ strictly convex:

$$\mu (f(x)) > f(\mu (x))\mathrm{\hspace{20mm}(1)}$$

Conversely if $-f(x)$ is strictly convex: 

$$\mu (f(x)) < f(\mu (x))$$

An analogous property of the median has been presented ([Merkle et al 2005][2], [pdf][3]).

**Motivation**

I have a nonlinear [function][4] [(pdf)][5] of positive random variables, too complex to post here, not directly pertinent to this question; I am looking for a more general answer. It is worth noting that it is, however, neither strictly concave nor convex.

In practice, I find that the function of the medians provides a much better estimate of the median of the function than does the estimate of the mean of the function from the function of the means. I am interested in learning the conditions for which this is true.

**Question** 

Under what conditions will the function of a median be closer to the median of a function than the mean of a function is to a function of the mean?

Specifically for what types of $f(x)$ and $x$ is

$$|\mu (f(x)) - f(\mu (x))| > |m (f(x)) - f(m (x))|$$


**References**

[Merkle et al 2005 Jensen's inequality for medians. Statistics & Probability Letters, Volume 71, Issue 3, 1 March 2005, Pages 277-281][6]


  [1]: http://stats.stackexchange.com/q/4655/1381
  [2]: http://dx.doi.org/10.1016/j.spl.2004.11.010
  [3]: http://milanmerkle.com/documents/radovi/SPL-71.pdf
  [4]: http://www.esajournals.org/doi/full/10.1890/0012-9615%282001%29071%5B0557%3AAMFSVD%5D2.0.CO%3B2
  [5]: http://www.oeb.harvard.edu/faculty/moorcroft/publications/publications/Moorcroft_etal_01.pdf
  [6]: http://dx.doi.org/10.1016/j.spl.2004.11.010