I actually got bored with typical questions in the forum. This thread interesting.

I worked 5 years in analytics. I developed statistical model for few BIG firms in US. Some of them part of Basel II norms. I have used probit, logit, GAM,...etc.

Let me start from Dason. I am not against probit. My experience was both probit and logit were giving similar results( KS, concordace,., etc). But there are cases probit has performed better than logit. The difference was very less.

One main reason i liked logit because the estimation is straight forward ( not a black box). Have you tried to estimated standared error of beta estimates using probit? The logit SE estimates similar to regression and this gives lot of simplicity.

Using latent variable approach we could explain index models. Instead of probit use,CDF of skewed normal( i guess this make lot of sense). By assuming other CDF as index function we assume different sigmoid curve. If one can understand probit they will understand this too.

Now if we're thinking about purely academic thought experiments I wonder how well the poisson inverse cdf would do as the link function for an integer valued covariate in the binomial situation...

I had mention the

**support is R** in the first post. Other range is also possible to assume. for Eg. Uniform. This is linear probability model ( but probability can go outside the (0,1) range).

There are some R&D type analytics companies experiment models with different alternatives. They explore the academic literature. But in general in practice, I found interpretation is one of the key elements in model development.