Correcting the Negativity of High-Order Kernel Density Estimators

Two methods are suggested for removing the problem of negativity of high-order kernel density estimators. It is shown that, provided the underlying density has at least moderately light tails, each method has the same asymptotic integrated squared error (ISE) as the original kernel estimator. For example, if the tails of the density decrease like a power of |x|−1, as |x| increases, then a necessary and sufficient condition for ISEs to be asymptotically equivalent is that a moment of order 1 + ϵ be finite for some ϵ > 0. The important practical conclusion to be drawn from these results is that in most circumstances, the bandwidth of the original kernel estimator may be used to good effect in the new, nonnegative estimator. A numerical study verifies that this is indeed the case, for a variety of different distributions.