Abstract
Recently, a new set of fundamental stereological formulae based on isotropically oriented probes through fixed points have been derived, the so-called “nucleator” estimation principle (cf. Jensen and Gundersen (1989, J. Microsc., 153, 249–267)). In the present paper, it is shown how a model-based version of these formulae leads to stereological estimators of reduced moment measures of stationary and isotropic random sets in ℝn.
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References
Ambartzumian, R. V. (1981). Stereology of random planar segment processes, Rend. Sem. Mat. Univ. Politec. Torino, 39, 147–159.
Baddeley, A., Gundersen, H. J. G. and Cruz-Orive, L. M. (1986). Estimation of surface area from vertical sections, J. Microsc., 142, 259–276.
Diggle, P. J. (1983). Statistical Analysis of Point Processes, Academic Press, London.
Federer, H. (1969). Geometric Measure Theory, Springer, Berlin-New York.
Gundersen, H. J. G. (1988). The nucleator, J. Microsc., 151, 3–21.
Hanisch, K.-H. (1983). Reduction of n-th moment measures and the special case of the third moment measure of stationary and isotropic planar point processes, Math. Operationsforsch. u. Statist. Ser. Statist., 14, 421–435.
Hanisch, K.-H. (1985). On the second order analysis of stationary and isotropic planar fibre processes by a line intercept method, Geobild'85, 141–146.
Hanisch, K.-H. and Stoyan, D. (1981). Stereological estimation of the radial distribution of centres of spheres. J. Microsc., 122, 131–141.
Jensen, E. B. (1990). Recent developments in the stereological analysis of particles, Research Report No. 181, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, (submitted to Ann. Inst. Statist. Math.).
Jensen, E. B. and Gundersen, H. J. G. (1989). Fundamental stereological formulae based on isotropically oriented probes through fixed points—with applications to particle analysis, Proceedings of the 4th International Conference in Stereology and Stochastic Geometry, J. Microsc., 153, 249–267.
Jensen, E. B. and Kiêu, K. (1990). A new integral geometric formula of Blaschke-Petkantschin type, Research Report No. 180, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus.
Mecke, J. (1967). Stationäre zufällige Masse auf Lokalkompakten Abelschen Gruppen, Z. Wahrsch. Verw. Gebiete, 9, 36–58.
Mecke, J. and Stoyan, D. (1980). Formulas for stationary planar fibre processes I—general theory, Math. Operationsforsch. Statist., Ser. Statist., 11, 267–279.
Nagel, W. (1987). An application of Crofton's formula to moment measures of random curvature measures, Forschungsergebnisse N/87/12, Friedrich-Schiller-Universität, Sektion Mathematik, Jena.
Schwandtke, A. (1988). Second-order quantities for stationary weighted fibre processes, Math. Nachr., 139, 321–334.
Simon, L. (1983). Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 3, 1–272.
Stoyan, D. (1981). On the second-order analysis of stationary planar fibre processes, Math. Nachr., 102, 189–199.
Stoyan, D. (1984). Further stereological formulae for spatial fibre processes, Math. Operationsforsch. u. Statist., Ser. Statist., 15, 421–428.
Stoyan, D. (1985a). Stereological determination of orientations, second-order quantities and correlations for random fibre systems, Biometrical J., 27, 411–425.
Stoyan, D. (1985b). Practicable methods for the determination of the pair-correlation function of fibre process, Geobild'85, 131–140.
Stoyan, D. and Ohser, J. (1982). Correlations between planar random structures, with an ecological application, Biometrical J., 24, 631–647.
Stoyan, D. and Ohser, J. (1985). Cross-correlation measures of weighted random measures and their estimation, Theory Probab. Appl., 29, 345–355.
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications, Akademie-Verlag, Berlin.
Zähle, M. (1982). Random processes of Hausdorff rectifiable closed sets, Math. Nachr., 108, 49–72.
Zähle, M. (1990). A kinematic formula and moment measures of random sets, Math. Nachr. (to appear).
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Jensen, E.B., Kiêu, K. & Gundersen, H.J.G. On the stereological estimation of reduced moment measures. Ann Inst Stat Math 42, 445–461 (1990). https://doi.org/10.1007/BF00049301
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DOI: https://doi.org/10.1007/BF00049301