Abstract
In the paper are suggested new robust estimators of location and variance. It is proved that these estimators have a breakdown point one half.
The used method comes from a geometric median. In the first step it is shown that we can employ one half of observations and the estimate stays robust in the sense of the breakdown point.
In the second step we show that we can add even more observations which are in some sense close to the geometric median and still get robust results. The robustness is proved in both steps for a multidimensional case.
Since we can employ more observations and stay robust in the sense of the breakdown point, we enlarge the used information in comparison to other robust estimators like median and therefore get better results. We combine the advantage of the robust estimator and the classical mean.
Our estimators are compared by simulation study with classical estimators like mean, median or alpha windsorised estimator. The comparison is done for different distributions like normal, Cauchy or exponential.
We also consider the case, when the normal distribution is with some probability contaminated by another distribution. In the last section are shown some illustrations of the approach.