Asymmetric S-shaped function mapping interval $[0, 1]$ to interval $[0, 1]$ In the literature, is there any asymmetric $S$-shaped function that maps the interval $[0, 1]$ to interval $[0, 1]$?
Unfortunately I can't post figure so I just describe what I mean in text. The function I want should be monotonic increasing. In the meanwhile, for a real number $c < 0.5$, the function
$$0 \mapsto 0, \quad c \mapsto 0.5, \quad 1 \mapsto 1.$$
Moreover, the function is smooth on its domain, concave on interval $[0, c]$ and convex on interval $[c, 1]$.
By the way, can it be seen as a sort of link-function?
 A: Yes.
And here is how you go about finding one.
For our purposes, convex means $F''(x)\ge 0$ and concave means $F''(x)\le 0$.
Ok, so let $F$ be such a function. If we also assume monotonicity, we have $F'(x)\ge 0$, and $F$ is a cumulative distribution function. Therefore, the convex and concave conditions are $f'(x)\ge 0$ for $x\le c$ and $f'(x) \le 0$ for $x\ge c$ (where $f=F'$ is the pdf of $F$).
In other words, now we are looking for density function on $[0,1]$ that is increasing on $[0,c]$ and decreasing on $[c,1]$. We go to a table of probability distributions on $[0,1]$ (e.g., Wikipedia's list) and see that the cumulative distribution function of the Beta distribution fits the bill.
Another approach would be to explicitly construct such a function (I believe a quartic would do the job).
A: I would comment instead, but I don't have the score yet, so here's my two cents:
The only thing I could think of is an asymmetric tangent function, yet I couldn't find anything about them except for this part of the Proceedings of the Estonian Academy of Sciences, Engineering. See if it helps...?
