but your procedure is a bit like a Cauchy principal value
Interestingly, we can imagine doing the integral of G (the inverse of the CDF) that you define. The Cauchy principal value is like integrating G between x- and x+ such that G(x-)=-y and G(x+)=y, and letting y go to infinity. The averaging I described is like integrating G between x and 1-x and letting x tend to zero.
Interestingly, we can imagine doing the integral of G (the inverse of the CDF) that you define. The Cauchy principal value is like integrating G between x- and x+ such that G(x-)=-y and G(x+)=y, and letting y go to infinity. The averaging I described is like integrating G between x and 1-x and letting x tend to zero.