Error analysis for general multtvariate kernel estimators

Kernel estimators for d dimensional data are usually parametrized by either a single smoothing parameter, or d smoothing parameters corresponding to each of the coordinate directions. A generalization of each of these parameterizations is to use a d× d matrix which allows smoothing in arbitrary directions. We demonstrate that, at this level of generality, the usual error approximations and their numerical minimization can be done quite simply using matrix algebra. The minimization formulas have the practical importance that they can be applied to data-driven selection of the smoothing parameters using a ”plug-in approach. Particular attention is paid to the special case of kernel estimation of multivariate normal mixture densities where it is shown that the numerical evaluation and minimization of both asymptotic and exact mean integrated squared error can be set up in a matrix algebraic formulation which requires no numerical integration. This provides a flexible family of multivariate smoothing problems for which error analyses can be performed in a computationally simple manner.