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Some quasi-arithmetic means of random variables easily give unbiased stronglyconsistent closed-form estimators of the joint of the location and scaleparameters of the Cauchy distribution. The one-step estimators of thosequasi-arithmetic means of the Cauchy distribution are considered. We establishthe Bahadur efficiency of the maximum likelihood estimator and the one-stepestimators. We also show that the rate of the convergence of the mean-squarederrors achieves the Cramer-Rao bound. Our results are also applicable to thecircular Cauchy distribution.

Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution