Mp. Wand et Mc. Jones, COMPARISON OF SMOOTHING PARAMETERIZATIONS IN BIVARIATE KERNEL DENSITY-ESTIMATION, Journal of the American Statistical Association, 88(422), 1993, pp. 520-528
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.