How to transform correlated random variables into uncorrelated ones (Q1585546)

For an arbitrary random vector \({\mathbf X}= (X_1,X_2, \dots, X_n)\), we can always construct uncorrelated random variables \(Y_1,Y_2, \dots, Y_n\) and \((R\to R)\) functions \(f_1,f_2, \dots,f_n\), such that \[ (X_1, X_2, \dots, X_n)= (f_1(Y_1),f_2(Y_2), \dots, f_n(Y_n)). \] Although the \(f\) s cannot always be one-to-one, in many important cases, the \(f\) s are not only one-to-one but also piecewise linear, e.g. if \({\mathbf X}\) is normally distributed. (This way, in many statistical models, the nuisance parameters can easily be transformed, such that their MLEs become uncorrelated with other parameters).