Comparison of three methods for determining fit parameter uncertainties for the Marquardt Compromise

Monte Carlo simulation was used to study the fit parameter uncertainties obtained from the Marquardt Compromise for a noisy data set, data that deviates from the parent function. A known Gaussian was the parent function in this work. Each data point was also weighted by either statistical or uniform weights. The three uncertainties were the Monte Carlo estimate, equal to the standard deviation of a fit parameter, the error matrix estimate, and the propagation of errors estimate. All simulations produced equality of the error matrix and the propagation of errors estimates for the same fit parameter. Each Monte Carlo estimate was linearly dependent on a parameter that determined the amount of noise and was independent of the weights. However, the error matrix uncertainties depended linearly on the weights and were independent of the noise for statistical weights, but depended linearly on the noise and were independent of the weights for fixed weights. Because the error matrix estimates depend on chi square, it was also studied and varied quadratically with the noise and as the inverse square of the weights. Each of these results was also derived from the theory of the Marquardt Compromise.