Geometric approach to statistical analysis on the simplex

The geometric interpretation of the expected value and the variance in real Euclidean space is used as a starting point to introduce metric counterparts on an arbitrary finite dimensional Hilbert space. This approach allows us to define general reasonable properties for estimators of parameters, like metric unbiasedness and minimum metric variance, resulting in a useful tool to better understand the logratio approach to the statistical analysis of compositional data, who's natural sample space is the simplex.