  A new method of bandwidth selection for kernel density estimators is
proposed. The method, termed indirect cross-validation, or ICV, makes use of
so-called selection kernels. Least squares cross-validation (LSCV) is used to
select the bandwidth of a selection-kernel estimator, and this bandwidth is
appropriately rescaled for use in a Gaussian kernel estimator. The proposed
selection kernels are linear combinations of two Gaussian kernels, and need not
be unimodal or positive. Theory is developed showing that the relative error of
ICV bandwidths can converge to 0 at a rate of $n^{-1/4}$, which is
substantially better than the $n^{-1/10}$ rate of LSCV. Interestingly, the
selection kernels that are best for purposes of bandwidth selection are very
poor if used to actually estimate the density function. This property appears
to be part of the larger and well-documented paradox to the effect that "the
harder the estimation problem, the better cross-validation performs." The ICV
method uniformly outperforms LSCV in a simulation study, a real data example,
and a simulated example in which bandwidths are chosen locally.
