A note on superkernel density estimators

It is well known that so-called superkernel density estimators have better asymptotic properties than conventional kernel estimators (and generally finite-order estimators) in the case when the density to be estimated is very smooth. In this note, we study asymptotic behavior of the mean integrated square error of superkernel density estimators in the case when the density to be estimated is not very smooth. It turns out that in this case, superkernel estimators still have better asymptotics than finite-order estimators.