Significance of Difference between Two Extremes (Maxima/Minima) I am comparing two different populations (with different sizes) of models, for each individual I have a score. I'd like to compare not only the difference of the two means but the maximum value. I've learnt in the past to perform significance test when comparing averages between groups but this is the first time that I'm trying to do the same with an extreme value (maximum in this case). To summarise, my question is: what is the right way to know whether the difference  $max(P_1) - max(P_2)$ is significant? 
 A: There may be a little confusion about the intricacies of inferential statistics and significance testing.
Inferential statistics do not check whether averages of samples differ between groups. We test whether the underlying distributions are different, at least as far as we can tell based on the samples.
It just happens that for the normal distribution, the sample mean is a good estimator of one of the distributional parameters. If we assume, e.g., equal variances, then we can use the sample means to make inferences 
However, the normal distribution (which I assume you are assuming) is unbounded. It has no maximum. Therefore, it really makes no sense in an inferential framework to ask whether maxima of samples from two groups differ differ significantly as maxima.
You could in principle work along the lines of "the maximum of this sample of size 100 is 1000, and the maximum of that sample of size 200 is 0.3, so it is unlikely they come from the same normal distribution". This page lists properties of maxima of finite samples of the normal distribution, e.g., the standard deviation of the sample maximum. However, this does not address the problem of estimating the parameters of the normal distribution from the data. (Note that bootstrapping will not work for maxima without additional effort.)
