An important factor to consider is where to do you want to live? It would be more difficult to get a job in the west with a PhD from Asia than one from a school in the west, even if the Asian school was more selective. However, stay away from from for-profit schools or schools that are not fully accredited in the U.S.
As far as research is concerned it does not matter from where did you do your research,what matters is quality of your research. If you do your research from any top or bottom university in west and if you are not able to publish in any good rated journal then your research is of no value,similarly if you do your research from any Asian university and you are able to publish in a good rated journal that gives credibility to your research . However it is a perception in Asia that if you do research in any third grade university in west it is better than any first grade university in east. This is prejudice or stereotyping which is difficult to change.