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Variance prediction under systematic sampling with geometric probes

  • Ximo Gual Arnau (a1) and Luis M. Cruz-Orive (a2)

In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝ n by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

Corresponding author
Postal address: Departament de Matematiques, Universitat Jaume I, Campus Riu Sec, E-12071 Castellón, Spain.
∗∗ Postal address: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain. Email address:
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Research supported by the Fundació Caixa Castelló grant no. P1A-94-24 and the Dirección General de Investigación Cientifica y Técnica Project No. #PB94-1058.

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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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