I would like to test two relatively small samples against the null hypothesis that both their means and variances are the same. The alternative would be that they in fact differ. I saw a post on this site advocating a ML test but I recall there is also a named test for this case, which I would ideally like to use in R, but whose name I forget. Can anyone help? Normality assumptions might be reasonable but would be difficult to test.
I would argue that it isn't possible to properly perform a joint test on the first two moments without knowing more about the distribution. Since there is no general rule as to how moments interact, it is impossible to construct a tight confidence region.
If you dare to make a normality assumption on both samples, the situation changes since the first two moments define the normal distribution, such that for example the Kolmogorov-Smirnoff test is equivalent to testing the first two moments. Be aware though that you would be using asymptotic results on assumed distributions.