Confidence Interval of CDF I am trying to determine if there is a statistically meaningful distinction between the cumulative probability density curves shown in the figure below.  

It's simple enough to do a $t$-test on the means of these distributions. But I am also looking to see if the treatment has an effect at more extreme values of the density distribution. For instance, if the means are the same but the 85th percentiles are different, that is something I would be interested in.
The 95% confidence interval of the mean is roughly $\bar{x} \pm 1.95 \sigma_x$. But it doesn't feel right to use the same variance at every level of the CDF, especially when the empirical distribution is largely non-normal.
 A: Assuming that your curves represent the empirical CDFs obtained from data, the usual way to test for a difference between more than two groups would be some kind of multi-sample non-parametric test akin to the Kolmogorov-Smirnov test, or a rank-based ANOVA test like the multi-sample Kruskal-Wallis test.  There are a number of papers in the statistical literature looking at multi-sample non-parameetric tests of this kind (see e.g., Kiefer 1959, Birnbaum and Hall 1960, Conover 1965, Sen 1973 for early literature).  If you reduce down to a pairwise comparison of interest, you can of course use the traditional two-sample tests.
There is an R package called ksamples that implements the multi-sample Kruskal-Wallis test and some other multi-sample non-parametric tests.  I am not aware of a package that does the multi-sample KS test, but others may be able to point you to additional resources.
A: For comparing 2 distributions at a time ("pairwise"), it's possible to find all the ranges of values for which the CDFs are statistically significantly different, while controlling the familywise error rate (FWER) at your desired level.  This (new) approach is described in detail in this 2018 Journal of Econometrics paper, as well as in this 2019 Stata Journal article.  R and Stata code (and open drafts of articles, and replication files) are at https://faculty.missouri.edu/~kaplandm.  Both articles include examples with real data.  Everything is fully nonparametric, and the "strong control" of FWER is exact even in small samples.
