Comparing predictions from models I'm wondering how to compare the predictions of three different models - a logit, a probit and a linear probability model - when predicting a binary outcome. I'm currently working with simulated data, so I have access to both the latent variable $y$ and the binary outcome $Y$ with $\Pr(Y=1)=y$
I'm under the impression that normal tools for model selection (i.e. AIC) do no perform very well in this setting due to the difference in the assumptions of the models. 
AUC or $corr(y,\hat{y})$ is a possibility, but I am wondering if this accurately reflects that the difference between models should be most evident in the tails of the distribution. 
I hope the above makes my bewilderment clear, otherwise please feel free to ask for clarifications in the comments.
EDITED FOR CLARIFICATION: I am performing some simulations because I'm curious about the performance of various predictors. One often hears, for example, that the difference between the logit and probit model is in the thickness of the tails. I was merely wondering what would be the best statistic for comparing the fit of the predictions $y$ to the real values $\hat{y}$. Since I am comparing two nonlinear models with a linear model, the various forms of $R^2$ is ill-suited. Hope that helps!
Best,
Andreas
 A: Still don't have a good grasp of what you're looking for. But if you're comparing binary classification models I wouldn't recommend just using AUC (a measure of discrimination)            
Usually one uses three metrics together, choosing one from each of the three categories below. :
(1) Global measure:  scaled Brier or N's R-sq. The difference between Brier and N's R-sq may be significant due to different penalty functions
The Brier score is often preferred, but should usually be converted to a scale Brier score (thought this is less of an issue of all analysis on same dataset).
(2) Discrimination: AUC, discrimination slope
(3) Calibration:  calibration slope with calibration or validation graph                   
A: Personally, I would use probit or logit regression, because LPM could estimate numbers that are not probability. 
Firstly look at your scatter plot, because in that graph you could see which model fits the data better. 
Then, if you need an estimate, you could try to use a Pseudo-R2. There are a wide variety of pseudo-R-square statistics as you can see here: http://www.ats.ucla.edu/stat/mult_pkg/faq/general/Psuedo_RSquareds.htm.
