Estimate dependence in medium dimensions, using ranks and sub-sampling

It is well known that non-parametric methods suffer from the "curse of dimensionality". We propose here a new estimation method for a multivariate distribution, using sub-sampling and ranks, which seems not to suffer from this "curse". We prove that in case of independence, the uncertainty of the estimated distribution increases almost linearly w.r.t. the dimension, for dimensions around 6. Otherwise, a simulation study shows that if we use this estimation to build an independence test, the number of observations needed to obtain a given power increases linearly with the dimension. Finally, we give examples of a regression using this estimation: with 3000 observations, in dimension 5, with a markedly complicated dependence, the estimated distribution is graphically very similar to the real one.