Let's say we have the a set $(X_i,Y_i)$, $i\in I$, $I$ is an arbitray finite set of indexes, and the model
$$ Y = g(X\beta)$$
Using some method, we obtain the individual first four moments of the residuals $u_i = Y_i - g(X_i, \hat{\beta})$, so we end up with the following set
$$ (Y_i, X_i, u_i, m^i_1, m^i_2, m^i_3, m^i_4)$$
Now comes my question, if I select a residual $u_j$, $j\in I$, and I have a diferent set of moments $(\hat{m}^k_1, \hat{m}^k_2, \hat{m}^k_3, \hat{m}^k_4)$, $k \not \in I$. How can I transform the residual $u_j$ to a residual $u_k$ equivalent in the sense if I had a diferent distribution for the residuals in each point, with those sets of four moments, the quantile of the residual would be the same? Alternatively, could I find a transformation (linear if possible) from the moments of $u_j$ to the moments of $u_k$?
