High-efficiency and robust M-estimates of the scale parameter on the Q-estimate basis

The commonly employed highly efficient and robust Q-estimate of the scale parameter proposed by Rousseeuw and Croux has been approximated using computationally fast Huber M-estimates. The suggested M-estimates were shown to be robust and highly efficient for an arbitrary underlying data distribution due to correctly choosing the approximation parameters. The following indicators of the efficiency and robustness of M-estimates of scale were computed: their asymptotic variances, influence functions and breakdown points. Special attention was given to the particular cases of the Gaussian and Cauchy distributions. It is noteworthy that for the Cauchy distribution, the suggested robust estimate of scale coincides with the maximal likelihood estimate. Finally, the computation time of these highly efficient and robust estimates of scale is 3-4 times less than for the corresponding Q-estimates.