Concerning the choice of the link function in binomial regression (e.g. logit versus probit or cauchit), I wonder what the recommended comparison criterion might be. Note that I am not interested in hypothesis testing, but just in a relative criterion for "goodness".
Obvious candidates for comparing link functions are the AIC, the Brier score, or the test statistic of the Hosmer-Lemeshow test, but the latter depends on the somewhat arbitrary choice of binning quantiles.
Is one of these recommended, or are they even equivalent? Are there other suggestion?
I would suspect, e.g., that AIC and Brier score are asymptotically equivalent, because the Brier score is a measure for prediction accuracy and the AIC is asymptotically equivalent to using leave-one-out MSE as a criterion.