Spearman coefficient for functions

We present a notion of Spearman's coefficient for functional data that extends the classical bivariate concept to situations where the observed data are curves generated by a stochastic process. Since Spearman's coefficient for bivariate samples is based on the natural data ordering in dimension one, we need to consider a data order in the functional context where a natural order between functions does not exist. The development uses a pre-order inspired in a depth definition but considering a down-up ordering instead of a center-outwards ordering of the sample. We show some of the main characteristics of Spearman's coefficient for functions and propose an independence test with a bootstrap methodology. We illustrate the performance of the new coefficient with both simulated and real data