How easy is a given density to estimate ?

In data analytic applications of density estimation one is usually interested in estimating the density over its support. However, common estimators such as the basic kernel estimator use a single smoothing parameter over the whole of the support. While this will be adequate for some densities there will be other densities that will be very difficult to estimate using this approach. The purpose of this article is to quantify how easy a particular density is to estimate using a global smoothing parameter. By considering the asymptotic expected L, error we obtain a scale invariant functional that is useful for measuring degree of estimation difficulty. Implications for the transformation kernel density estimators, which attempt to overcome the inadequacy of the basic kernel estimator, are also discussed.