ROBUST M-ESTIMATORS' Parameter Scale is Highly Efficient on Base Q Grades

Russiva and Crookes proposed the widely used, highly efficient, robust Q-score parameter scale in their 1993 work, and they approximated it using "fast" Huber's M-scores. Shown What proposed us M-scores are highly efficient and robust on arbitrary nom distributions, thanks to correct choice parameter approximations. We pay particular attention to the Cauchy divisions  and cases of Gaussian distribution . Important terms: Cauchy's law, parameter scale, robustness, and Gaussian distribution With the Q-estimate as a foundation, the M-estimates of the scale parameter are very efficient and robust. Rousseeuw and Croux (1993) proposed the scale parameter of Q-estimate, which is often used with fast Huber M-estimates, and we were able to get rather close to it. By meticulously choosing the approximation parameters, we proved that the proposed M-estimates are efficient and resilient for any data distribution. In order to measure the robustness and efficiency of scale M estimates , we calculated their asymptotic variances; breakdown points , and influence functions. The Cauchy and Gaussian distributions were our main focus. Notably, the proposed robust estimate is consistent the Maximum likelihood estimation for the Cauchy distribution. Last but not least, these robust and efficient scale estimates need three to four times less computing time than their comparable Q-estimates. The research provides robust and efficient solutions for estimating distribution parameters more quickly and with less complexity, making them suitable for applications requiring fast and accurate computations.