Distribution-free consistency of kernel non-parametric M-estimators

We prove that in the case of independent and identically distributed random vectors (Xi,Yi) a class of kernel type M-estimators is universally and strongly consistent for conditional M-functionals. The term universal means that the strong consistency holds for all joint probability distributions of (X,Y). The conditional M-functional minimizes (2.2) for almost every x. In the case M(y)=y the conditional M-functional coincides with the L1-functional and with the conditional median.