Existence and Approximation of Densities of Chord Length- and Cross Section Area Distributions
DOI:
https://doi.org/10.5566/ias.2923Keywords:
absolute continuity, chord length distribution, cross section area distribution, stereologyAbstract
In various stereological problems a n-dimensional convex body is intersected with an (n−1)-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the (n−1)-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.
References
Baddeley A, Jensen EBV (2004). Stereology for Statisticians. Boca Raton: Chapman and Hall/CRC.
Coleman R (1969), Random paths through convex bodies. J Appl Prob 6:430–441.
Davy P, Miles RE (1977). Sampling Theory for Opaque Spatial Specimens. J R Stat Soc Ser B Methodol 39:56–65.
Gardner RJ (2002). The Brunn-Minkowski inequality. Bull Amer Math Soc 39:355–405.
Gates J (1987). Some properties of chord length distributions. J Appl Prob 24:863–874.
Harutyunyan HS, Ohanyan VK (2009). The chord length distribution function for regular polygons. Adv Appl Prob 41:358–366.
van der Jagt T, Jongbloed G, Vittorietti M (2023). Stereological determination of particle size distributions for similar convex bodies. arXiv preprint arXiv:2305.02856
Klee V (1959). Some new results on smoothness and rotundity in normed linear spaces. Math Ann 139:51–63.
Koldobsky A (2005). Fourier Analysis in Convex Geometry. Providence: American Mathematical Society.
Miles RE (1978). The importance of proper model specification in stereology. In: Miles RE, Serra J, eds. Geometrical Probability and Biological Structures: Buffon’s 200th Anniversary. Berlin and New York: Springer-Verlag. 115–136.
Ohser J, Mu ̈cklich F (2000). Statistical Analysis of Microstructures in Materials Science. Chichester: John Wiley & Sons, Inc.
Paul JL (1981). Distribution curves of sectional areas through some families of convex particles. J Microsc 122:165–172.
Santalo ́ LA, Kac M (2004). Integral Geometry and Geometric Probability. Cambridge: Cambridge University Press.
Schneider R (2013). Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press.
Schuster EF (1985). Incorporating support constraints into nonparametric estimators of densities. Commun Statist Theory Methods 14:1123–1136.
Sheather SJ, Jones MC (1991). A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation. J R Stat Soc Ser B Methodol 53:683–690.
Wicksell SD (1925). The Corpuscle Problem. A Mathematical Study of a Biometric Problem. Biometrika 17:84–99.
Zamfirescu T (1987). Nearly all convex bodies are smooth and strictly convex. Monatsh Math 103:57–62.
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Copyright (c) 2023 Thomas van der Jagt, Geurt Jongbloed, Martina Vittorietti
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