The Robustness and Super-Robustness of L^p Estimation, when p < 1

Abstract: In robust statistics, the breakdown point of an estimator is the percentage of outliers with which an estimator still generates reliable estimation. The upper bound of breakdown point is 50%, which means it is not possible to generate reliable estimation with more than half outliers. In this paper, it is shown that for majority of experiences, when the outliers exceed 50%, but if they are distributed randomly enough, it is still possible to generate a reliable estimation from minority good observations. The phenomenal of that the breakdown point is larger than 50% is named as super robustness. And, in this paper, a robust estimator is called strict robust if it generates a perfect estimation when all the good observations are perfect. More specifically, the super robustness of the maximum likelihood estimator of the exponential power distribution, or L^p estimation, where p<1, is investigated. This paper starts with proving that L^p (p<1) is a strict robust location estimator. Further, it is proved that L^p (p < 1)has the property of strict super-robustness on translation, rotation, scaling transformation and robustness on Euclidean transform.