  In this paper we show how to use Fourier transform methods to analyze the
asymptotic behavior of kernel distribution function estimators. Exact
expressions for the mean integrated squared error in terms of the
characteristic function of the distribution and the Fourier transform of the
kernel are employed to obtain the limit value of the optimal bandwidth sequence
in its greatest generality. The assumptions in our results are mild enough so
that they are applicable when the kernel used in the estimator is a
superkernel, or even the sinc kernel, and this allows to extract some
interesting consequences, as the existence of a class of distributions for
which the kernel estimator achieves a first-order improvement in efficiency
over the empirical distribution function.
