Models for binary response that appear like logit/probit regression but never approach 1 I'm reading the textbook "Introduction to Econometrics" by Stock and Watson. Chapter 11 discusses regression with a binary response and it teaches the logit/probit models mostly. These seem to make sense when the plots show an S shaped relationship between X and Y, but what if I'm working with a dataset where the model calculates a probability in the shape of a "hill".
For example what if my continuous predictor X ranges from 0-100 and my response Y is only equal to 1 for X's at values of 30-70 but gets more concentrated at X's around 50. So lets say around X=50 we could have a 40% probability of Y=1, and then at X=40, X=60 we'd have 25% probability of Y=1 until the probability of Y=1 dissipates around 30-70. I cant find any models that model such a relationship through google - Are there any models that are more appropriate than logit/probit for this type of relationship?

The logit/probit function is the cdf of the conditional mean. So by definition, in a linear function of some variable, say x, the probability must approach 1 for some value.
Now, you can add some sort of non-linearity into your model of the conditional mean. That will allow you to have a "hill". Note that it will not truly be a hill, rather it will be an S-shaped curve in a curved space, so that when you "flatten" out the curved space and look at it from that "flat" perspective, it doesn't look S-shaped any more. A very simple and fairly standard way to do this is to add polynomial terms to your conditional mean. At that point, you are relying on the fact that you can create an arbitrary Taylor-expansion of any function.
Generally, we don't see people do that in an inferential setting as this complicates the marginal interpretation of the coefficients, which is already very messy with discrete choice models due to the non-linearities from the probability distribution.
