  Variable kernel density estimation allows the approximation of a probability
density by the mean of differently stretched and rotated kernels centered at
given sampling points $y_n\in\mathbb{R}^d,\ n=1,\dots,N$. Up to now, the choice
of the corresponding bandwidth matrices $h_n$ has relied mainly on asymptotic
arguments, like the minimization of the asymptotic mean integrated squared
error (AMISE), which work well for large numbers of sampling points. However,
in practice, one is often confronted with small to moderately sized sample sets
far below the asymptotic regime, which highly restricts the usability of such
methods. As an alternative to this asymptotic reasoning we suggest an axiomatic
approach which guarantees invariance of the density estimate under linear
transformations of the original density (and the sampling points) as well as
under splitting of the density into several `well-separated' parts. In order to
still ensure proper asymptotic behavior of the estimate, we \emph{postulate}
the typical dependence $h_n\propto N^{-1/(d+4)}$. Further, we derive a new
bandwidths selection rule which satisfies these axioms and performs
considerably better than conventional ones in an artificially intricate
two-dimensional example as well as in a real life example.
