I have read that the kernel intensity estimator of a point process is not consistent, because the variance at any point is "of order 1".
What does he mean by "of order 1"? I would like to understand why the variance does not go to 0.
Then (the kernel intensity estimator) essentially uses local information around the point of interest alone to obtain an estimate for the intensity. By smoothing locally, the resulting estimator will not be overly biased if the true intensity function is spatially continuous. But because the number of events in any fixed region is of order 1, the variance of the estimator does not diminish. As a result, (the kernel intensity estimator) is not a consistent estimator for the true intensity. [Guan]
[guan]: Yongtao Guan (2008) On Consistent Nonparametric Intensity Estimation for Inhomogeneous Spatial Point Processes, Journal of the American Statistical Association, 103:483, 1238-1247, DOI: 10.1198/016214508000000526