Multi-variate two-sample ANDERSON-DARLING TEST Suppose we receive a reading of samples taken from a sensor system with multiple variables, which may be assumed as continuous real values.
After few days, we receive another reading of samples from the same system.
How could we tell if both sets of samples came from the same distribution or there have been any differences?

*

*I understand that KS Test could spot the difference between the two samples.

*I also understand that ANDERSON-DARLING TEST is an alternate for this. According to some sources, which I have read, it is also applicable to k-samples and you don't necessarily need to know the distributions.

Question:
I want to use ANDERSON-DARLING TEST, for some reasons as listed here:
http://www.jaqm.ro/issues/volume-6,issue-3/pdfs/1_engmann_cousineau.pdf
What I don't get is, I have multiple variables. As per my understanding, these tests will run against one variable taken from previous data, and one variable from current data and comparison will be drawn. Is there a multi-variate approach for this? OR I will have to compare both datasets by taking one variable at a time. How will I reach a conclusion in that case?
Tx

The extreme amount of ties makes me skeptical of any of this, even if you go one-by-one through the marginal distributions. I would use something designed for binary tests. If you want to go one-by-one through the marginals, there are many good tests of proportions; my favorite right now is the G-test, but you might have one you prefer (e.g., chi-squared test).
Something nice about the G-test is that it gives somewhat obvious generalizations. What the G-test does is compare nested logistic regression models, one of which has the group variable and the other of which just has an intercept, much as ANOVA compares nested linear models where one model has the group variable and the other just has an intercept. The G-test then does a likelihood ratio test of the nested models.
You can go likelihood ratio tests of lots of different kinds of nested models.
The likelihood ratio test that I have in mind involves nested multivariate probit regressions. The simple model just has an intercept term, while the more complex model has an intercept term in addition to the group variable (same as the G-test or ANOVA). You then do a likelihood ratio test of the nested models.
