Say we have $n$ integers, $x_{1}\leq x_{2}\leq\ldots\leq x_{n}$. For each $i\in\{1,\ldots,n\}$, we can find a unique integer $h_{i}$ as the "largest $h$ satisfying there are at least $h$ of $x_{j}$ such that $x_{i}-x_{j}\geq h$". Rigorously, we define $$h_{i}=\mathop{\arg\max}_{h\in\{0,1,\ldots,n\}}\min\{h,x_{i}-x_{h}\}.$$ For example, for a set of integers $\{0,1,2,3,4,5,100\}$, the corresponding "H-indeces" is given by $\{0, 1, 1, 2, 2, 3, 6\}$; for $\{-1, 0, 0, 0, 0, 5, 100\}$ the corresponding H-indeces would be $\{0, 1, 1, 1, 1, 5, 6\}$.
It seems $h$ contains some information about the concentration level of $x_{i}$, and is not quite sensitive to the extreme values. I wonder does this $h$ have certain statistical implication? For fixed $n$, assume $X_{i}$ follows i.i.d. discrete distribution $P_{X}$. Is it possible to find some connection between the distribution of $h(X_{(1)}),\ldots,h(X_{(n)})$ and $P_{X}$? Any comments or suggestions are welcome.