Robustness of confidence intervals for scale parameters based on m-estimators

The robustness of confidence intervals for a scale parameter based on M-esimators is studied, especially in small size samples. The coverage probablity is used as measure of robustness. A theorem for a lower bound of the minimum coverage probability of M-estimators is presented and it is applied in order to examine the behavior of the standard deviation and the median absolute deviation, as interval estimators. This bound can confirm the robustness of any other scale M-estimator in interval estimation. The idea of stretching is used to formulate the family of distributions that are considered as underlying. Critical values for the confidence interval are computed where it is needed, that is for the median absolute deviation in the Normal, Uniform and Cauchy distribution and for the standard deviation in the Uniform and Cauchy distribution. Simulation results have been achieved for the estimation of the coverage probabilities and the critical values.