Consider two sets of data points A and B. Both these data points are from mixture of unknown number of Gaussians. The mean of the Gaussians are little different for each set (there may have few overlap or very close separated mean values). However for both cases the variance of all the Gaussians are small. Now, if we give a set of data point say C, how to estimate C is from A or from from B? I understand there are many methods to do so: is there a way tell the most efficient method? This is a very board question, so specifically can we compare the KS test https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test and https://en.wikipedia.org/wiki/Wasserstein_metric for this problem? Is there a way to prove that KS test/Wasserstein metric would give better estimate?
It appears to me that Cumulative distribution is not smooth so Wasserstein metric would be better, is it true?