I am looking for a function with some specific properties (this is for a probabilistic simulation). It should be a function that is runs though (and is symmetric) the origin and asymptotically approaches $-1$ and $1$ as its parameter goes from negative to positive. Any sigmoid function would fit the bill but I ideally want it to have a convex component around the zero (i.e. I want its value to be more similar around zero) and all sigmoid functions that I am aware of change rather quickly. See the attached badly drawn picture for an illustration.
Would appreciate any pointers! Computationally less expensive functions are prefered.
Best Answer
Try this: $$\frac{2}{\pi}\arctan(x^3)$$ The idea is if you have a sigmoid function that is linear around $0$, you can get the desired result if you change $x$ into something that varies slower than $x$ around $0$. If you want to change the width, replace $x^3$ with $\alpha x^3$.
The same would also work for $$\frac{2}{1+e^{-x^3}}-1$$