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A discrepancy in the determination of the proton's charge radius, $r_p$,between muonic hydrogen spectroscopy versus classic atomic spectroscopy andelectron scattering data has become known as the proton radius puzzle.Extractions of $r_p$ from electron scattering data require determination of theslope of the proton's charge form factor, $G_E^p$, in the limit of$Q^2\rightarrow0$ through fitting and extrapolation. Some works have presentedthe $Z$-transformation fitting technique as the best choice for this type ofextraction due to the true functional form of $G_E^p$ being mathematicallyguaranteed to exist within the parameter-space of the fit function. In thiswork, we test this claim by examining the mathematical bias and variancesintroduced by this technique as compared to the more traditional $Q^2$ fitsusing statistically sampled $G_E^p$ parameterizations with known input radii.Our tests conclude that the quality of the $Z$-transformation technique dependson the range of data used. In the case of new experiments, the fit function andtechnique should be selected in advance by generating realistic pseudodata andassessing the power of different techniques.

Direct Comparison of using a Z-Transformation instead of the traditional $Q^2$ for for Extraction of the Proton Radius from $e-p$ Scattering Data