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Variance prediction for pseudosystematic sampling on the sphere

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


Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.


Corresponding author

Postal address: Departament de Matemàtiques, Universitat Jaume I, Campus Riu Sec, E-12071 Castellón, Spain.
∗∗ Postal address: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avenida Los Castros s/n, E-39005 Santander, Spain. Email address:


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Research supported by the Ministerio de Ciencia y Tecnologia (Spain) I+D project BSA2001-0803-C02.



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[1] Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York.
[2] Cressie, N. A. C. (1991). Statistics for Spatial Data. John Wiley, New York.
[3] Cruz-Orive, L. M. (1989). On the precision of systematic sampling: a review of Matheron's transitive methods. J. Microscopy 153, 315333.
[4] Foley, A., Lane, D. A., Nielson, G. M., Franke, R. and Hagen, H. (1990). Interpolation of scattered data on closed surfaces. Comput. Aided Geometric Design 7, 303312.
[5] García-Fiñana, M. and Cruz-Orive, L. M. (2000). New approximations for the efficiency of Cavalieri sampling. J. Microscopy 199, 224238.
[6] Gual-Arnau, X. and Cruz-Orive, L. M. (2000). Systematic sampling on the circle and on the sphere. Adv. Appl. Prob. 32, 628647.
[7] Gundersen, H. J. G. (1988). The nucleator. J. Microscopy 151, 321.
[8] Jensen, E. B. V. (1998). Local Stereology. World Scientific, Singapore.
[9] Kiêu, K., Souchet, S. and Istas, J. (1999). Precision of systematic sampling and transitive methods. J. Statist. Planning Infer. 77, 263279.
[10] Lockwood, E. H. and Macmillan, R. H. (1978). Geometric Symmetry. Cambridge University Press.
[11] Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications (Cahiers Centre Morphologie Math. 5). École des Mines de Paris, Fontainebleau.
[12] Riordan, J. (1968). Combinatorial Identities. John Wiley, New York.
[13] Schlather, M. (1999). An introduction to positive definite functions and to unconditional simulation of random fields. Tech. Rep. ST 99–10, Lancaster University.
[14] Tandrup, T. (1993). A method for unbiased and efficient estimation of number and mean volume of specified neuron subtypes in rat dorsal root ganglion. J. Comparative Neurol. 329, 269276.


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Variance prediction for pseudosystematic sampling on the sphere

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


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