  The kernel estimator is known not to be adequate for estimating the density
of a positive random variable X. The main reason is the well-known boundary
bias problems that it suffers from, but also its poor behaviour in the long
right tail that such a density typically exhibits. A natural approach to this
problem is to first estimate the density of the logarithm of X, and obtaining
an estimate of the density of X using standard results on functions of random
variables (`back-transformation'). Although intuitive, the basic application of
this idea yields very poor results, as was documented earlier in the
literature. In this paper, the main reason for this underachievement is
identified, and an easy fix is suggested. It is demonstrated that combining the
transformation with local likelihood density estimation methods produces very
good estimators of R+-supported densities, not only close to the boundary, but
also in the right tail. The asymptotic properties of the proposed `local
likelihood transformation kernel density estimators' are derived for a generic
transformation, not only for the logarithm, which allows one to consider other
transformations as well. One of them, called the `probex' transformation, is
given more focus. Finally, the excellent behaviour of those estimators in
practice is evidenced through a comprehensive simulation study and the analysis
of several real data sets. A nice consequence of articulating the method around
local-likelihood estimation is that the resulting density estimates are
typically smooth and visually pleasant, without oversmoothing important
features of the underlying density.
