an:03936299
Zbl 0584.62140
Diggle, Peter
A kernel method for smoothing point process data
EN
J. R. Stat. Soc., Ser. C 34, 138-147 (1985).
00150828
1985
j
62M09 62G05 93E14 60G55 62M20
local intensity; correction for end-effects; density estimation; smoothing; kernel; Cox process; heterogeneity; selection of the bandwidth; mean-square-error; reduced second moment measure
Let \((x_ 1,...,x_ n)\) be a sample of a point process observed on (0,T). The author considers smoothing of these data by a kernel in the following way
\[
{\hat \lambda}_ t(x)=\sum^{n}_{i=1}\delta (| x-x_ i| /t)/\int^{T}_{0}\delta (| x-u| /t)du
\]
where \(\delta\) is a symmetric density. In the case of a Cox process \({\hat \lambda}{}_ t(x)\) is an estimate of the underlying realization of the rate process. It is not quite clear what \({\hat \lambda}{}_ t(x)\) estimates for a general point process, but it can serve as a useful tool for the explanatory analysis of the heterogeneity of an observed process.
The paper gives a procedure for the selection of the bandwidth t. In the case of a Cox process the mean-square-error \(M(t)=E(({\hat \lambda}_ t(x)-\lambda (x))^ 2)\) can be expressed for uniform \(\delta\) in terms of the intensity \(\mu\) and the reduced second moment measure K. The procedure selects t by minimizing \(\hat M(\)t) obtained from M(t) by substituting the usual estimators \({\hat \mu}\) and \(\hat K\) for \(\mu\) and K. The method is illustrated by simulated data and some real data on joints along a coal seam.
H.R.K??nsch