  The application of global non-analytical fit to the quantitative spectral analysis of the bilayer-split peak-dip-hump lineshape of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ at $(\pi,0)$ are revisited. The robustness of the results is verified by various fitting schemes confirming physically the correlation of the superconducting peak with the superfluid density. 
Robustness of the global non-analytical fit to the analysis of   peak-dip-hump lineshape in bilayer-split spectra of   Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$