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A kernel method for smoothing point process data. (English) Zbl 0584.62140
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.
Reviewer: H.R.Künsch

##### MSC:
 62M09 Non-Markovian processes: estimation 62G05 Nonparametric estimation 93E14 Data smoothing in stochastic control theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 62M20 Inference from stochastic processes and prediction
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