  Three different permutation test schemes are discussed and compared in the
context of the two-sample problem for functional data. One of the procedures
was essentially introduced by Lopez-Pintado and Romo (2009), using notions of
functional data depth to adapt the ideas originally proposed by Liu and Singh
(1993) for multivariate data. Of the new methods introduced here, one is also
based on functional data depths, but uses a different way (inspired by
Meta-Analysis) to assess the significance of the depth differences. The second
new method presented here adapts, to the functional data setting, the
k-nearest-neighbors statistic of Schilling (1986). The three methods are
compared among them and against the test of Horvath and Kokoszka (2012) in
simulated examples and real data. The comparison considers the performance of
the statistics in terms of statistical power and in terms of computational
cost.
