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

TL;DR

This paper investigates the efficiency of maximum likelihood and one-step estimators for quasi-arithmetic means of the Cauchy distribution, demonstrating their optimal convergence rates and applicability to circular Cauchy models.

Contribution

It establishes the Bahadur efficiency of these estimators and shows they achieve the Cramer-Rao bound, extending results to circular Cauchy distributions.

Findings

Bahadur efficiency of MLE and one-step estimators proven.

Convergence rate matches the Cramer-Rao bound.

Results applicable to circular Cauchy distribution.

Abstract

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.