Robust estimator design with an emphasis balance between performance and robustness

From a number of ML estimators (typically unbiased) of practical interest which include the variance for a Gaussian distribution, the standard deviation for a Laplace distribution, the variance for a Rayleigh distribution and a “spread parameter” for a Cauchy distribution, we design robust estimators according to an emphasis balance between normalized performance and normalized robustness. We measure performance with inverted MSE and robustness with a differential geometric approach.