Estimators for the Cauchy distribution

We discuss the properties of various estimators of the central position of the Cauchy distribution. The performance of these estimators is evaluated for a set of simulated experiments. Estimators based on the maximum and mean of the posterior probability density function are empirically found to be well behaved when more than two measurements are available. On the contrary, because of the infinite variance of the Cauchy distribution, the average of the measured positions is an extremely poor estimator of the central position. However, the median of the measured positions is well behaved. The rms errors for the various estimators are compared to the Fisher-Cramer-Rao lower bound. We find that the square root of the variance of the posterior density function is predictive of the rms error in the mean posterior estimator.