  Important information concerning a multivariate data set, such as clusters
and modal regions, is contained in the derivatives of the probability density
function. Despite this importance, nonparametric estimation of higher order
derivatives of the density functions have received only relatively scant
attention. Kernel estimators of density functions are widely used as they
exhibit excellent theoretical and practical properties, though their
generalization to density derivatives has progressed more slowly due to the
mathematical intractabilities encountered in the crucial problem of bandwidth
(or smoothing parameter) selection. This paper presents the first fully
automatic, data-based bandwidth selectors for multivariate kernel density
derivative estimators. This is achieved by synthesizing recent advances in
matrix analytic theory which allow mathematically and computationally tractable
representations of higher order derivatives of multivariate vector valued
functions. The theoretical asymptotic properties as well as the finite sample
behaviour of the proposed selectors are studied. {In addition, we explore in
detail the applications of the new data-driven methods for two other
statistical problems: clustering and bump hunting. The introduced techniques
are combined with the mean shift algorithm to develop novel automatic,
nonparametric clustering procedures which are shown to outperform mixture-model
cluster analysis and other recent nonparametric approaches in practice.
Furthermore, the advantage of the use of smoothing parameters designed for
density derivative estimation for feature significance analysis for bump
hunting is illustrated with a real data example.
