The Pitman estimator of the Cauchy location parameter

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This paper examines the estimation of the Cauchy location parameter when the scale parameter is known. Using the squared error loss function, a closed form of the minimum risk equivariant (MRE) estimator is derived. While several properties of the estimator are discussed, this article focuses particularly on the efficiency of the MRE (or the Pitman) estimator in finite samples. A simulation study indicates that the gain in efficiency when using the MRE estimator rather than the MLE is particularly important in small samples. We also compare the performance of the Pitman estimator with that of other equivariant estimators, including M- and L-estimators and some approximations to the Pitman estimator. In addition, we study the unconditional and conditional inference of the Cauchy location parameter based on both the MLE and the Pitman estimator. We further assess the small sample robustness properties of the estimators included in the simulation by contaminating the Cauchy distribution at three different levels of contamination.