I am trying to estimate a utility function on a given dataset. The data is normalized and ranges between 0 & 1 on both the $X$ & $Y$ axes.
The estimation will find the function that best fits the data. Theory specifies that the function must be concave down and increase.
Here is an image for reference:
On the left is what I am looking for.
The function can be steep and approach 1 on the Y-axis very quickly or the function can be much flatter. It just has to be concave down and increasing.
Are there any known equations with parameters that can be estimated using a minimization routine that is flexible enough to capture the desired shapes described above?