Estimation of integrated squared density derivatives

Kernel density estimators are used for the estimation of integrals of various squared derivatives of a probability density. Rates of convergence in mean squared error are calculated, which show that appropriate values of the smoothing parameter are much smaller than those for ordinary density estimation. The rate of convergence increases with stronger smoothness assumptions, however, unlike ordinary density estimation, the parametric rate of n-1 can be achieved even when only a finite amount of differentiability is assumed. The implications for data-driven bandwidth selection in ordinary density estimation are considered.