Infill asymptotics and bandwidth selection for kernel estimators of spatial intensity functions

We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We derive expansions for the bias and variance in the scenario that n independent copies of a point process in R^d are superposed. When the same bandwidth is used in all d dimensions, we show that an optimal bandwidth exists and is of the order n^−1/(d+4) under appropriate smoothness conditions on the true intensity function.