I'm currently a graduate student at UChicago (I live in Hyde Park), and I don't really feel that safety here is an issue. If you avoid bad neighborhoods, walking around at night, and taking the public transportation frequently, then I don't think there is much reason to worry. The level of safety in Chicago varies greatly depending which neighborhood you are in, and Hyde Park for example, where the University is located, is relatively safe with many professors living here. The south loop is also fairly safe also, and you can stay in these neighborhoods exclusively and basically experience all the university has to offer.
There was an international student who was killed last year in a mugging; I don't want to make light of that situation, or assign any blame to the victim at all, but I think I read that he refused to hand over his laptop (which he had out).
As far as your specific research interests, I can't really comment on which school is a better fit, however, I think if you are geared towards theoretical research in academia, UChicago might be a better fit. The stats courses are very rigorous and heavily focused on theory and so are the courses in other departments, but they still provide relevant practical experience. The general feel of the university is that, as a whole, it is aimed towards students who truly enjoy the subject and are pursuing academia (compare to state schools for example, which as a whole seem more aimed at sending students into positions in industry). I think this might be important, since you can take courses in other departments outside of statistics and they will still be generally good quality.
Also, in the US, UChicago is in general much more prestigious that CMU, with UChicago being on the same tier as ivy-league schools while CMU is not. Further, I have a strong suspicion that our placements for stats graduates interested in academia is probably better than CMU on average, but I'm not completely sure. On the other hand, CMU seems to be much better at computer science, so if you're interested in something related to that then that is something to consider. Although, despite the relatively low ranking, I still think the CS department here is pretty good, plus you can take courses/do research at TTIC.