Adaptive Smoothing and Density-Based Tests of Multivariate Normality

Methods of adaptive smoothing of density estimates, where the amount of smoothing applied varies according to local features of the underlying density, are investigated. The difficulties of applying Taylor series arguments in this context are explored. Simple properties of the estimates are investigated by numerical integration and compared with the fixed kernel approach. Optimal smoothing strategies, based on the multivariate Normal distribution, are derived. As an application of these techniques, two tests of multivariate Normality—one based on integrated squared error and one on entropy—are developed, and some power calculations are carried out.