I need to perform an analysis that predicts proportions that theoretically could range from 0 to positive infinity. I have two primary problems.

1. My data contains undefined values. My response variable is a proportion of two frequencies, but sometimes those proportions are undefined since the denominator frequency equals zero (e.g., 0.5 divided by 0).

2. My data contains zeros. The gamma distribution seems best suited for my data, but whenever the numerator frequency equals zero, the proportion equals zero as well. The gamma distribution can't handle zeros. Poisson and negative binomial aren't available since their outputs are restricted between 0 and 1, and I need to predict proportions that can be greater than 1.

I've considered using a hurdle model with the gamma distribution to deal with the zero inflation in my data, but I don't like that it can't differentiate true zeros from false zeros. A hurdle model also doesn't address my undefined data problem.

Any ideas or suggestions for either of these issues? Is there a distribution I'm not aware of that would be better?