How to deal with multi-modal distributions in hypothesis testing?

Say that I collected random variable X from one population and a random variable Y from another population.

I want to apply a statistical test to determine whether these two populations are different. However, I notice that X is multimodal (there are two peaks in the data). This makes it difficult to perform standard statistical testing methods. How would most researchers proceed from here?

• I don't think we can meaningfully answer this without knowing what hypothesis you are testing. For many hypotheses, the multimodality would have no effect on the test. For example, $H_0: X-Y = 0$ vs. $H_1: X-Y > 0$. What difference does it make if X or Y are multimodal? $H_0$ and $H_1$ are still perfectly valid hypotheses without consideration of the number of modes of the distributions. – David Marx Mar 14 '16 at 17:50
• Hm, a Student's t-test assumes a Gaussian for each of the populations and we could not apply it for multimodal distribution. – user46925 Mar 14 '16 at 17:55
• @DavidMarx: your comment is very confusing it looks like your are testing whether the difference of the random variables $X, Y$ are different. But we should be testing whether some measure of these random variables, such as the mean, is different. – Cliff AB Mar 14 '16 at 18:44
• If one population distribution is bimodal and the other is not, the distributions aren't the same. However, if you do want to test if two samples share a common distribution (rather than say compare means), one such test is the two-sample Kolmogorov-Smirnov test, which is discussed many times here. Please clarify your null and alternative hypotheses. – Glen_b Mar 15 '16 at 0:14
• What if it was a very small peak caused by noise? And they did indeed come from the same distributions – user46925 Mar 15 '16 at 0:35