  Copula modelling has become ubiquitous in modern statistics. Here, the
problem of nonparametrically estimating a copula density is addressed. Arguably
the most popular nonparametric density estimator, the kernel estimator is not
suitable for the unit-square-supported copula densities, mainly because it is
heavily affected by boundary bias issues. In addition, most common copulas
admit unbounded densities, and kernel methods are not consistent in that case.
In this paper, a kernel-type copula density estimator is proposed. It is based
on the idea of transforming the uniform marginals of the copula density into
normal distributions via the probit function, estimating the density in the
transformed domain, which can be accomplished without boundary problems, and
obtaining an estimate of the copula density through back-transformation.
Although natural, a raw application of this procedure was, however, seen not to
perform very well in the earlier literature. Here, it is shown that, if
combined with local likelihood density estimation methods, the idea yields very
good and easy to implement estimators, fixing boundary issues in a natural way
and able to cope with unbounded copula densities. The asymptotic properties of
the suggested estimators are derived, and a practical way of selecting the
crucially important smoothing parameters is devised. Finally, extensive
simulation studies and a real data analysis evidence their excellent
performance compared to their main competitors.
