# "Median" version of L-moments

The L-moments are useful as robust summary statistics for various probability distributions, similar to the moments but only requiring the mean of the distribution to exist. Each L-moment is a linear sum of expected values of order statistics. I had the thought that we could perhaps improve on this if we use the median instead, giving us a set of "M-moments," let's call them, which exist for all probability distributions, even if the mean doesn't exist.

We would like to take the definition of L-moment and replace the expected value $$\mathbf{E}X$$ with the median $$\textbf{M}X$$. The L-moment is a sum of expected values of order statistics, so when moving to the median there are two slightly different ways to do this: we can take a sum of medians or a median of sums, which give very slightly different results. Both seem reasonable and I'm not sure if either is "right," but perhaps one is well-studied. I'll call the two versions $$M_n$$ and $$N_n$$: \begin{align} M_1 &= \textbf{M}X & N_1 &= \textbf{M}X \\ M_2 &= (\textbf{M}X_{2:2} - \textbf{M}X_{1:2})/2 & N_2 &= \textbf{M}(X_{2:2} - X_{1:2})/2 \\ M_3 &= (\textbf{M}X_{3:3} - 2 \textbf{M}X_{2:3} + \textbf{M}X_{1:3})/3 & N_3 &= \textbf{M}(X_{3:3} - 2 X_{2:3} + \textbf{M}X_{1:3})/3 \\ M_4 &= (\textbf{M}X_{4:4} - 3 \textbf{M}X_{3:4} + 3 \textbf{M}X_{2:4} - \textbf{M}X_{1:4})/4 & N_4 &= \textbf{M}(X_{4:4} - 3 X_{3:4} + 3 X_{2:4} - X_{1:4})/4 \\ \cdots & & & \cdots \end{align}

where $$X_{k:n}$$ is the $$k$$'th order statistic from a random sample of size $$n$$ (treated as another random variable).

The question is: do these things have a name, and have they been studied? Do we know how robust, efficient, etc they are? It's such a simple and basic idea that it seems very likely this is well known, and it seems extremely useful that these exist for every probability distribution.