Probit transformation for kernel density estimation on the unit interval

Abstract

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.