  Kernel estimation of a probability density function supported on the unit
interval has proved difficult, because of the well known boundary bias issues a
conventional kernel density estimator would necessarily face in this situation.
Transforming the variable of interest into a variable whose density has
unconstrained support, estimating that density, and obtaining an estimate of
the density of the original variable through back-transformation, seems a
natural idea to easily get rid of the boundary problems. In practice, however,
a simple and efficient implementation of this methodology is far from
immediate, and the few attempts found in the literature have been reported not
to perform well. In this paper, the main reasons for this failure are
identified and an easy way to correct them is suggested. It turns out that
combining the transformation idea with local likelihood density estimation
produces viable density estimators, mostly free from boundary issues. Their
asymptotic properties are derived, and a practical cross-validation bandwidth
selection rule is devised. Extensive simulations demonstrate the excellent
performance of these estimators compared to their main competitors for a wide
range of density shapes. In fact, they turn out to be the best choice overall.
Finally, they are used to successfully estimate a density of non-standard shape
supported on $[0,1]$ from a small-size real data sample.
