This question is about handling zeros in an independent variable for a regression.
In particular, the zeros are not missing data or true zeros, but occur because of quantization. As a concrete example, lets say the observations are cities, and the variable is the number (or fraction) of people in some category, based on a sample. If the sample for a particular city is small, it might have zero people in a category, even if the true number in the city's population is nonzero.
In this case, what are possible ways to work with zeros if the variable is heavy tailed? Normally I would log-transform, but I can't do that when zeros are present, and because many of the observations are zero, excluding them would introduce a large bias.
Some things I'm considering: other transformations, replacing the variable with a bayesian estimate of the fraction, switching from regression to ANOVA with people as observations and city as a categorical variable. Are these valid approaches? Am I missing any? What are the pros and cons?