Resolving velocity distribution function parameters from observations with significant Poisson statistical uncertainty

Several analyses of particle observations aim to determine the distribution functions of physical parameters that characterize observed systems. Some standard analysis methods determine these distributions by fitting mathematical models to the data. The accuracy of the fitting techniques depends on the treatment of the observations and their uncertainties. Here, we evaluate the performance of three fitting techniques by applying them to simulated electron observations, which are governed by the Poisson distribution. We specifically examine and quantify the accuracy of two standard chi-squared minimization techniques and a maximum-likelihood method. The chi-squared minimization techniques simplify the analysis by treating the measurement uncertainties as Gaussian errors. Although such a simplification reduces the complexity of the calculations in some occasions, it may lead to systematic errors in the determined parameters. On the other hand, the maximum-likelihood method considers the exact Poisson probability for each data-point and returns accurate parameters for all the examples we examine here. We highlight the importance of using the appropriate method when the observations are accompanied by significant statistical uncertainty. Nevertheless, the methods we examine here, converge to the same answer as the statistical uncertainty of the observations reduces.