On Idempotent Estimators of Location

Idempotence is a well-known property of functionals of location. It means that the value of the functional at a singular distribution must be identically to the mass point of this distribution. First, we explain the role of idempotence in the known axiomizations of location functionals. Then we derive the distribution of idempotent and sufficient statistics. In the special cases of parametric families of location we get the so-called power-n-distributions. Power-n-distributions again are distributions with a parameter of location and can be derived from every location family for which the density is constrained. Additionally we show that the completeness of the populations family insures the completeness of the family of power-n-distributions. And at last, we give a further, now very easy proof that the normal distribution is the only one for which a idempotent, sufficient and unbiased estimator attains the Cramer-Rao-lower bound.