Explicit solutions for the asymptotically-optimal bandwidth in cross validation

We show that least squares cross-validation (CV) methods share a common structure which has an explicit asymptotic solution, when the chosen kernel is asymptotically separable in bandwidth and data. For density estimation with a multivariate Student t(ν) kernel, the CV criterion becomes asymptotically equivalent to a polynomial of only three terms. Our bandwidth formulae are simple and non-iterative (leading to very fast computations), their integrated squared-error dominates traditional CV implementations, they alleviate the notorious sample variability of CV, and overcome its breakdown in the case of repeated observations. We illustrate with univariate and bivariate applications, of density estimation and nonparametric regressions, to a large dataset of Michigan State University academic wages and experience.