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Resampling marked point processes. (English) Zbl 0946.62087
Ghosh, Subir (ed.), Multivariate analysis, design of experiments, and survey sampling. A tribute to Jagdish N. Srivastava. New York, NY: Marcel Dekker. Stat., Textb. Monogr. 159, 163-185 (1999).
From the introduction: Suppose $$\{X({\mathbf t}), {\mathbf t}\in\mathbb{R}^d\}$$ is a homogeneous random field in $$d$$ dimensions, with $$d\in\mathbb{Z}^+$$, that is, a collection of real-valued random variables $$X({\mathbf t}$$) that are indexed by the continuous parameter $${\mathbf t}\in\mathbb{R}^d$$. In the important special case where $$d= 1$$, the random field $$\{X({\mathbf t})\}$$ is just a continuous time, stationary stochastic process. The probability law of the random field $$\{X({\mathbf t}), {\mathbf t}\in\mathbb{R}^d\}$$ will be denoted by $$P_X$$. We will generally assume that $$EX({\mathbf t})^2< \infty$$, in which case homogeneity (i.e., strict stationarity) implies weak stationarity, namely that for any $${\mathbf t},{\mathbf h}\in\mathbb{R}^d$$, $$EX({\mathbf t})= \mu$$, and $$\text{Cov}(X({\mathbf t}), X({\mathbf t}+{\mathbf h}))= R({\mathbf h})$$; in other words, $$EX({\mathbf t})$$ and $$\text{Cov}(X({\mathbf x}), X({\mathbf t}+{\mathbf h}))$$ do not depend on $${\mathbf t}$$ at all.
Our objective is statistical inference pertaining to features of the unknown probability law $$P_X$$ on the basis of data; in particular, this paper will focus on estimation of the common mean $$\mu$$. For the case where the data are of the form $$\{X({\mathbf t}), {\mathbf t}\in{\mathbf E}\}$$, with $${\mathbf E}$$ being a finite subset of the rectangular lattice $$\mathbb{Z}^d$$, different block-resampling techniques have been developed in the literature.
Our objective will be interval estimation of $$\mu$$ on the basis of measurements of the value of $$X(\cdot)$$ at a finite number of generally non-lattice, irregularly spaced points $${\mathbf t}\in\mathbb{R}^d$$. The observed marked point process is then defined as the collection of pairs $$\{({\mathbf t}_j, X({\mathbf t}_j))$$, $$j= 1,\dots, N(K)\}$$, where $$\{{\mathbf t}_j\}$$ are the points at which the $$\{X({\mathbf t}_j)\}$$ “marks” happen to be observed.
The paper is organized as follows: Section 2 contains some useful notions on mixing, and some necessary background on mean estimation, in Section 3 the marked point process “circular” bootstrap is introduced and studied, while in Section 4 the marked point process “block” bootstrap is introduced and studied; some concluding remarks are presented in Section 5, while all proofs are deferred to Section 6.
For the entire collection see [Zbl 0927.00053].

##### MSC:
 62M40 Random fields; image analysis 62F40 Bootstrap, jackknife and other resampling methods