Two sample hypothesis tests for functional (curve) data Suppose I have two samples of functional data or curves $x_i(t), i \in \{1,\dots,n_1\}$ and $y_i(t)\in\{1,\dots,n_2\}$ for $t\in\{1,\dots,T\}$. What tests are suitable for testing the null hypothesis that $x_i$ and $y_i$ come from the same distribution?

You have a model for your data, some linear combination of basis functions.
Now do that again (including the intercept), but times an indicator variable for one of the groups.
Now you consider the first model to be nested within the model with the interactions. F-test the two models to get your p-value.
This is a natural extension to FDA of an ANCOVA with an interaction. In effect, you are asking if you can explain $Y$ better by allowing the two groups to have different functional representations.
