COMPARISON OF SMOOTHING PARAMETERIZATIONS IN BIVARIATE KERNEL DENSITY-ESTIMATION

The basic kernel density estimator in one dimension has a single smoot hing parameter, usually referred to as the bandwidth. For higher dimen sions, however, there are several options for smoothing parameterizati on of the kernel estimator. For the bivariate case, there can be betwe en one and three independent smoothing parameters in the estimator, wh ich leads to a flexibility versus complexity trade-off when using this estimator in practice. In this article the performances of the differ ent possible smoothing parameterizations are compared, using both the asymptotic and exact mean integrated squared error. Our results show t hat it is important to have independent smoothing parameters for each of the coordinate directions. Although this is enough for many situati ons, for densities with high amounts of curvature in directions differ ent to those of the coordinate axes, substantial gains can be made by allowing the kernel mass to have arbitrary orientations. The ''spherin g'' approaches to choosing this orientation are shown to be detrimenta l in general, however.