Maximum size prediction in Wicksell’s corpuscle problem for the exponential tail data. (English) Zbl 1037.62098

A collection of random spheres in \(R^3\) is considered. The centres of the spheres form a Poisson point process, the area of a great circle of the sphere is independent of its position and has a CDF \(F_V(x)\). \(F_A(x)\) is the CDF of the area of circles which are planar sections of the spheres cut by a plane. The objective of the paper is to predict the characteristics of the upper tail of \(1-F_V(x)\) by observations from \(F_A\). It is proposed to use a truncated exponential model \[ F_V(x)= {\mathbb I}\{0\leq t\leq u\}(1-\exp(-(t-u)/\xi)) \] under which \[ (1-F_A(t))/(1-F_A(u))=\exp(-(t-u)/\xi),\quad t\geq u. \] Problems of the choice of a threshold value are discussed. Results of simulations are presented. This technique is applied to the analysis of the size of graphite nodules in spheroid graphite cast iron.


62M99 Inference from stochastic processes
62M30 Inference from spatial processes
60D05 Geometric probability and stochastic geometry
62P30 Applications of statistics in engineering and industry; control charts
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