Marginal distribution fitting and copula I have simulated many pseudo-observations from a nonparametric copula density estimate (for that I used a bootstrap approach). I now want to go back to the original space, but I can't use any inversion strategy since I don't know my margins (I have estimated them also in a nonparametric way, using Kernel density estimators).
 A: We can simply treat this as a univariate problem -- you're taking generated uniform values $u_i$ and for a given cdf $F$, you need to figure out what quantiles $x_i$ have $F(x_i) = u_i$.
A kernel density estimate is a mixture of the kernel, centered at each data value, with bandwidth related to the standard deviation of the kernel in a predefined way (differently for different kernels - for example for the Gaussian kernel, the bandwidth is the standard deviation of the kernel)
So the cdf, $F$, will be a mixture of the corresponding component cdfs. 
Now you need to "invert" the cdf, but you don't literally need to produce a functional form, you just need to find the $x_i$ in the equation above. This could be done numerically.
For example, you could start by using linear interpolation based on the positions of the original observations. A given pseudo-value $x_i=F^{-1}(u_i)$ will lie between two of them -- call them $y_{(t)}$ and $y_{(t+1)}$, say --  unless it's outside the extreme values, so you can identify $v_{(t)}=F(y_{(t)})$ and similarly $v_{(t+1)}$ being the quantiles of the transformed-to-uniform original values that lie either side of $u_i$, then use linear interpolation to get an approximation to $x_i$), and then update that first approximation using root finding methods (note that you have the derivative of the cdf -- it's the original kernel). For the cases beyond the extremes you could do (for example) do root finding starting at the extremes, though for given kernels you may be able to identify better starting positions.
This won't be particularly fast (though there are ways it could be sped up), but it can be done.
