Inference for thinned point processes, with application to Cox processes (Q1069630)

Let \((N_ 1,...,N_ n)\) be i.i.d. copies of a point process on an arbitrary compact space \(E\) and let \((N'_ 1,...,N'_ n)\) be thinnings of \((N_ 1,...,N_ n)\) with rate \(p(x)\). This means that a point \(x\in N_ i\) is deleted independently of the other ones with probability \(1-p(x)\). The paper considers inference about \(p(x)\) and about the distribution of N based on the observation of \(N'_ 1,...,N'_ n\). Let \((A_{n1},...,A_{nk_ n})\) be a partition of E. The estimate of \(p(x)\) is then \[ \hat p(x)=\sum^{n}_{i=1}N'_ i(A_{nj})/(nE(N(A_{nj})))\text{ if }x\in A_{nj}. \] If the partition gets finer with increasing n in a suitable way, \(\hat p(x)\) is shown to be consistent and asymptotically normal. With \(\hat p(x)\) and the empirical Laplace functional of \(N'\) a consistent estimate of the Laplace functional of \(N\) is obtained. Further results include an asymptotically consistent test whether \(N'\) is indeed a thinning of \(N\) with some rate and a reconstruction of \(N\) from \(N'\) in the case of a Cox process. Unfortunately no example with simulated or real data illustrates the potential applicability of the results.