All doubts are originated from my interests in multi-modality of globular clusters. Astronomers used luminosity functions (LF), which is in general represented by a histogram because of binning, and they visually identified multi-modality to explain multiple epochs of star formation history of a galaxy. Then, later Ashman, Bird and Zepf (1994) introduced the likelihood approach to resolve/prove the bimodality problem statistically. However, as Protossov, et. al. (2002) pointed out, there are regularity conditions such as finite expectation and identifiability, and I’ve seen papers on globular clusters using likelihood ratio tests (LRT) for the hypothesis testing of say, 2 generations of clusters vs 3 generations of clusters with gaussian mixture models, where these regularity conditions are violated and LRT cannot be applied for such a hypothesis testing.
Not all statistics are robust until someone proved its robustness case by case. The part that annoys me about the chi-square is out of sudden, researchers said, “due to chi-square…” What if the conditions on those chi-square methods do not satisfy the nature of the data set as some astronomers used LRT where it cannot be applied for the hypothesis testing on their data sets but already applied to other data sets of the same object type satisfying the mathematical conditions? Because of its fame and convenience, for me at least (most?), people tend to use chi-square blindly.

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