  Some quasi-arithmetic means of random variables easily give unbiased strongly
consistent closed-form estimators of the joint of the location and scale
parameters of the Cauchy distribution. The one-step estimators of those
quasi-arithmetic means of the Cauchy distribution are considered. We establish
the Bahadur efficiency of the maximum likelihood estimator and the one-step
estimators. We also show that the rate of the convergence of the mean-squared
errors achieves the Cramer-Rao bound. Our results are also applicable to the
circular Cauchy distribution.
