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Local block bootstrap for inhomogeneous Poisson marked point processes. (English) Zbl 1381.60091

The literature is vast with resampling methods for data observed at regularly spaced points in time and space, see [S. N. Lahiri, Resampling methods for dependent data. New York, NY: Springer (2003; Zbl 1028.62002)]. The most common assumption for the occurrence of irregularly spaced data is that of a Poisson process. Another central assumption is that the locations of points are independent of the associated measurements, also called “marks”, at these locations. Let \(\{X(t): t\in \mathbb R^d\}\) be a real-valued, strictly stationary random field in the continuous, \(d\)-dimensional parameter \(t\); Let \(\{N(t): t\in\mathbb R^d\}\) be an inhomogeneous Poisson point process with rate \(\lambda(t)\). The point process \(N(t)\) is assumed to be independent of the random field \(X(t)\). Let \(\tau_1\), \(\tau_2\),…, \(\tau_{N(K)}\) denote the points generated by \(N(t)\) inside the observation region \(K\) (a compact, convex subset of \(\mathbb R^d\)). The pairs \(\{X(\tau_i), N(\tau_i)\}\) for \(i=1,\dots, N(K)\) constitute an observed data from an inhomogeneous marked point process. Consider the problem of estimation the mean \(\mu= \operatorname{E}X(t)\) based on the above marked point process data. A natural estimation is the sample mean \({{1}\over{N(K)}} \int_K X(t) N(dt)\). Note that the denominator includes a random quantity \(N(K)\) which causes difficulties. Denote \({\bar{X}}_K = {{1}\over{N(K)}} \sum_{i=1}^{N(K)} X(\tau_i)\) and \({\widetilde{X}}_K = {{1}\over{\Lambda(K)}} \sum_{i=1}^{N(K)} X(\tau_i)\). Note that \({\widetilde{X}}_K\) is not a proper statistic unless \(\Lambda(K)\) is known which is unrealistic since the rate \(\lambda(t)\) is typically unknown. The sample mean \({\bar{X}}_K\) is the statistic of choice. For the longest time, it was thought that \({\bar{X}}_K\) and \({\widetilde{X}}_K\) are asymptotically equivalent. However, this is only true when \(\mu=0\), see [the authors, J. Appl. Probab. 50, No. 3, 889–892 (2013; Zbl 1274.60064)]. Still, the asymptotic normality of \({\bar{X}}_K\) can be inferred from the asymptotic normality of \({\widetilde{X}}_K\) although the asymptotic variance are different. This is achieved in Section 2 under standard moment and mixing conditions. In Section 3, the authors show how existing methods for resampling homogeneous marked point processes can be adapted to the inhomogeneous case when \(d=1\). Section 4 is devoted to the local block bootstrap procedure for general \(d\)-dimensional inhomogeneous marked point processes data and its validity for the sample mean and related statistics are established. Finally, Section 5 is devoted to the comparison of the finite-sample performance for these methods. Technical proofs are given in the appendix.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
62G09 Nonparametric statistical resampling methods
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