A counterexample to a conjecture on the bandwidth selection criterion of Cronie and Van Lieshout

Cronie and Van Lieshout (2018) introduced a nonparametric bandwidth selector for kernel estimation of spatial point process intensity functions. Using the Gaussian kernel and no edge correction term in their criterion, they conjectured that the associated statistic $T_\mathcal{K}(h;\psi,W)$ is a monotone function of the bandwidth $h$. We show that there exist point patterns $\psi$ for which the derivative of this statistic is strictly negative at a positive bandwidth $h$, which, because the statistic tends to zero for $h \to 0$ and to infinity for $h \to \infty$, disproves the conjecture. Such point patterns consist of a sufficiently large cluster of tightly packed points together with a single well-separated point.

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