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A bootstrap algorithm for the isotropic random sphere. (English) Zbl 0831.62075

Summary: Let \(\{X_{\vec p}\}\) be a real-valued, homogeneous, and isotropic random field indexed in \(\mathbb{R}^3\). When restricted to those indices \(\vec p\) with \(|\vec p|\), the Euclidean length of \(\vec p\), equal to \(r\) (a positive constant), then the random field \(\{X_{\vec p}\}\) resides on the surface of a sphere of radius \(r\). Using the author’s modified stratified spherical sampling plan \({\mathcal P}\) on the sphere [A finite sampling plan, central limit theorem, and bootstrap algorithm for a homogeneous and isotropic random field on the 3-dimensional sphere. Ph.D. Thesis, Dpt. Stat., Univ. North Carolina/Chapel Hill, Mimeo Ser. 2097 (1993)], define \(\{X_{\vec p}\): \(\vec p\in {\mathcal P}\}\) to be a realization of the random process and \(|{\mathcal P}|\) to be the cardinality of \({\mathcal P}\).
A bootstrap algorithm is presented and conditions for strong uniform consistency of the bootstrap cumulative distribution function of the standardized sample mean, \(|{\mathcal P} |^{-1/2} \sum_{\vec p\in {\mathcal P}} (X_{\vec p}- {\mathbf E} (X_{\vec p}\})\), are given. We illustrate the bootstrap algorithm with global land-area data.

MSC:

62M40 Random fields; image analysis
62G09 Nonparametric statistical resampling methods
60G60 Random fields
62G20 Asymptotic properties of nonparametric inference
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