  We investigate the asymptotic mean squared error of kernel estimators of the
intensity function of a spatial point process. We show that when $n$
independent copies of a point process in $\mathbb R^d$ are superposed, the
optimal bandwidth $h_n$ is of the order $n^{-1/(d+4)}$ under appropriate
smoothness conditions on the kernel and true intensity function. We apply the
Abramson principle to define adaptive kernel estimators and show that
asymptotically the optimal adaptive bandwidth is of the order $n^{-1/(d+8)}$
under appropriate smoothness conditions.
