Which link function for a regression when Y is continuous between 0 and 1? I've always used logistic regression when Y was categorical data 0 or 1.
Now I have this dependent variable that is really a ratio/probability. That means it can be any number between 0 and 1. 
I really think "logistic" shape would fit very nicely, but I remember categorical Y was a big deal when proving why MLE works.
The point is, I am wrong using logit regression for this Y or it doesn't matter? Should I use probit instead?  
Am I committing a capital crime?
 A: There's nothing wrong per se with using "logistic regression" for this kind of data. You can think of it as an empirical adjustment to allow fitting a response that has a bounded support. It's better than the alternative (logit-transforming your response, then using ordinary linear regression) because the resulting predictions are asymptotically unbiased, the mean predicted value equals the observed mean response, and (probably the most important) you don't have to worry about situations where Y equals 0 or 1. The arcsin transformation can handle Y = 0 or 1, but then your regression results aren't so easily interpretable in terms of log-odds ratios.
The main thing to look out for is that, as with any generalized linear model, you are implicitly assuming a particular relationship between the $E(Y|X)$ and $\textrm{Var}(Y|X)$. You should check that this assumption holds, eg by looking at diagnostic plots of residuals.
For most cases, doing a probit regression will give very similar results to a logistic regression. An alternative is to use the complementary-log-log link if you have reason to believe there is asymmetry between Y = 0 and 1.
A: Link functions convert the expected value of Y (given X) to something that is unbounded.  While in logistic regression, Y takes values 0 or 1, the logit isn't applied to Y but to Pr(Y=1|X).  (The logit of 0 and 1 are each undefined.)  So it's perfectly reasonable to use the logit or the probit in this case.
The other thing to think about is the residual variance: is there a particular transformation that would best stabilize the variance for your case?  For proportions, the arcsine square-root transformation is often used, as it is variance-stabilizing for binomial proportions.  Consider the discussion here.
