Physical Sciences

The $Z$-transformation fitting technique purports to be the best choice for this type of extraction due to the true functional form of $G_E^p$ being mathematically guaranteed to exist within the parameter-space of the fit function. In this article, we examine the mathematical bias and variances introduced by choosing this technique as compared to the more traditional $Q^2$ fits to directly test if it is truly a better technique. A selection of $G_E^p$ parameterizations with known fit radius were statistically sampled and fit with in both $Q^2$ and in $Z$. The mean and variance of the extracted radii were compared to the input radius. The fits were performed both with the parameters unbound and bound to study the effect of choosing physically-motivated bounds. Additionally, the data are fit with Rational(N,M) functions in $Q^2$ as used by the PRad collaboration. As expected, the results in both $Q^2$ and $Z$ were poor for unbounded polynomials, with the fits in $Q^2$ showing a rather large bias with small variance while the fits in $Z$ had a small bias with a large variance. The application of bounds yield small improvements. In most cases, fits in $Q^2$ with Rational(N,M) type functions yielded improved results. We find that the $Z$-transformation technique is a useful tool, but not a universal improvement over fitting in $Q^2$. The "best" technique was dependent on the parameterization being fit and the $Q^2$ range of the data. The fits with a Rational(N,M) function in $Q^2$ typically provided improved results over other techniques when the data has sufficient $Q^2$ range, but provided unphysical results when the range was too small. We conclude that any planned fit should make use of psuedo data to determine the best basis and functional form for a given data set. In the case of new experiments, this selection should ideally be done prior to the collection of the data.