Iterated Transformation–Kernel Density Estimation

Abstract

Transformation from a parametric family can improve the performance of kernel density estimation. In this article we give two data-driven estimators for the optimal transformation parameter. We demonstrate that multiple families of transformations can be used at the same time, and there can be benefits to iterating this process. The transformation scheme can be expected to first pick the right transformation family and then pick the optimal parameter. Insight as to the performance of the method comes from our analysis of a number of real datasets, two of which are included in this article. To illustrate the effectiveness and asymptotics of the transformation method, we also present results on one of the five target densities used in our simulation study. It is then proved that the Johnson family of transformations, when coupled with transformation-kernel density estimation, makes a wide variety of density shapes easier to estimate. The transformation method has overall better performance than the usual method and in many cases is much more effective.