Let $F$ be a continuous c.d.f that is strictly increasing on $[a,b]$,
where $a=\inf{x: F(x)>0}$ and
$b=\sup{x: F(x)<1}$.< p="">A control function is any function
$\kappa:[0,1]\to[c,d]\subseteq[a,b]$ that
is continuous and strictly increasing
and which fulfills $\kappa(0)=c$ and $\kappa(1)=d$.
The $\kappa$-index of the distribution $F$ (Gagolewski, Grzegorzewski, 2010)
is a number $\rho_\kappa\in(0,1)$
such that
$$\rho_\kappa=1-F(\kappa(\rho_\kappa)).$$
It turns out that under certain conditions in a model of i.i.d. random variables
the S-statistic associated with $\kappa$ is an asymptotically
unbiased, normal and strongly consistent estimator of $\rho_\kappa$.