This paper deals with the convergence in Mallows metric of classical multivari-ate kernel distribution function estimators. Moreover, in order to obtain a better fitting of the unknown distribution function* F, a new locally orientated kernel smooth estimator, which also converges to F in Mallows metric, is introduced. It follows the consistency of smoothed bootstrap for regular functions of the vector mean, which could improve classical bootstrap performances. Two simple simulation studies show the improvements of the smoothed versions of the bootstrap with respect to the classical one.