Which link function in binomial regression is better? 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.

I think it should be a matter of interpretation of model results.  The logit link has you modeling the log odds, or equivalently the multiplicative effect in the odds for a one-unit increase in a covariate.  The probit link has you modeling the standard normal percentile, or equivalently, the additive effect in the standard normal percentile for a one-unit increase in a covariate.
From my experience as uncomfortable as odds and odds ratios can be they seem much more tangible and immediately relevant than standard normal percentiles unless the original endpoint was in fact a continuous normally distributed endpoint that was dichotomized for the purposes of binary regression.  In that case the probit link would be relevant to talk in terms of percentiles or standard deviations from the mean.  Likewise for the Cauchit link if the original data are in fact Cauchy distributed.
