Let f be a probability density and C be an interval on which f is
bounded away from zero. By establishing the limiting distribution of the uniform
error of the kernel estimates fn of f, Bickel and Rosenblatt (1973) provide con-
dence bands Bn for f on C with asymptotic level 1 .. 2]0; 1[. Each of the
con dence intervals whose union gives Bn has an asymptotic level equal to one;
pointwise moderate deviations principles allow to prove that all these intervals
share the same logarithmic asymptotic level. Now, as soon as both pointwise and
uniform moderate deviations principles for fn exist, they share the same asymptotics.
Taking this observation as a starting point, we present a new approach for
the construction of con dence bands for f, based on the use of moderate deviations
principles. The advantages of this approach are the following: (i) it enables to
construct con dence bands, which have the same width (or even a smaller width)
as the con dence bands provided by Bickel and Rosenblatt (1973),but which have
a better aymptotic level; (ii) any con dence band constructed in that way shares
the same logarithmic asymptotic level as all the con dence intervals, which make
up this con dence band; (iii) it allows to deal with all the dimensions in the same
way; (iv) it enables to sort out the problem of providing con dence bands for f
on compact sets on which f vanishes (or on all Rd), by introducing a truncating
operation.