Question

How can we prove the similarity between two overlaid curves statistically?

Answer 1

1 - Compute the Fourier transform of both functions.

2 - Compare the absolute values of the transforms using a distance like Itakura or Itakura-Saito. There are others.

There is another interesting way that was used in speech recognition based on the Bellman mathematical programing (if my memory does not fail) . Essentially consisted on putting the values of the two functions in two axis. Then the Cartesian product was done. To go from one point to another there is a cost (may be the distance). The total minimum cost way gives a distance between the two curves and states the modification to go from one to the other. If they are equal, aside a translation, the way will be linear.