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

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