Local-Likelihood Transformation Kernel Density Estimation for Positive Random Variables

The kernel estimator is known not to be adequate for estimating the density of a positive random variable <i>X</i>. The main reason is the well-known boundary bias problems that it suffers from, but also its poor behavior 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 <i>X</i>, and obtaining an estimate of the density of <i>X</i> using standard results on functions of random variables (“back-transformation”). Although intuitive, the basic application of this idea usually yields poor results, as was documented earlier in the literature. In this article, 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 behavior of those estimators in practice is evidenced through a comprehensive simulation study and the analysis of several real datasets. 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.