Estimation of the intensity of a Poisson point process by means of nearest neighbor distances

Abstract

A new unbiased consistent asymptotically normal estimator Uk of the intensity λ of a stationary multivariate Poisson point process is exhibited. This estimate is based on a combination of the j‐th nearest neighbor (possibly non Euclidean) distances (j=1, ..., k) to a single fixed site x. A simple closed form containing logarithmic terms is obtained for E(Ulk)(0<l<k).

Infill Asymptotics and Bandwidth Selection for Kernel Estimators of Spatial Intensity Functions

“…For such n, by a change of variables and the symmetry of the Beta kernels, and writing ḡ for an upper bound to the pair correlation function, the integral in the last line in (7) can be bounded in absolute value by…”

“…Various non-parametric estimators are available to do so. Some techniques are based on local neighbourhoods of a point, expressed for example by its nearest neighbours [7], its Voronoi [11] or Delaunay tessellation [13,14]. By far the most popular technique, however, is kernel smoothing [6].…”

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“…Silverman (1986). In order to increase robustness against outliers, one may prefer to combine k-th nearest neighbour distances for several values of k (Granville 1998). Although Eqs.…”