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].

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].