Abstract:

Arithmetical mean and median usually serve as basic examples of M-estimators (Lehman 83). Both of them are L-estimators. Thus there is a natural question whether there are some other L-estimators which are M-estimators as well. We shall show that, with rare exceptions, this is not the case. More precisely, we shall show that the arithmetical mean and empirical quantiles are the only L-estimators with nonnegative coefficients having a nontrivial $\psi$-function.