State estimation for Cox processes with unknown probability law
Let \(N_ i\), \(i\geq 1\), be i.i.d. observable Cox processes on a compact metric space E, directed by unobservable random measures \(M_ i\) and assume that the probability law of the \(M_ i's\) is completely unknown. In this paper techniques are developed to estimate the conditional Laplace functional (LF) \[ (*)\quad f\to E(\exp \{-\int fdM_{n+1}\}| {\mathcal F}^{N_{n+1}}) \] of \(M_{n+1}\) using data from the processes \(N_ 1,N_ 2,...,N_ n\). To be more precise, the following results are proved: 1) Let M be a diffuse random measure on E with finite mean measure and let N be a Cox process directed by M. Then the conditional LF (*) \(f\to E(\exp \{-\int fdM\}| {\mathcal F}^ N)\) can be represented in terms of the LF \(L_ M(\mu,f)\) of the Palm processes \(M^{(\mu)}\) of M. 2) Let M be a diffuse random measure on E with finite mean measure and let N be a Cox process directed by M. Then for almost all finite and integer-valued measures \(\mu\) on E, the reduced Palm process \(N^{(\mu)}\) is a Cox process directed by the Palm process \(M^{(\mu)}\). Using the well known relation \(L_ M(\mu,f)=L_ N(\mu,- \log (1-f))\) this implies that the conditional LF (*) can be represented in terms of the LF \(L_ N(\mu,f)\) of the reduced Palm processes \(N^{(\mu)}\) of N. 3) Estimators of the LF \(L_ N(\mu,f)\) are given and it is shown that these estimators are strongly consistent and asymptotically normal.
strongly consistent and asymptotically normal
scientific article; zbMATH DE number 3925905