Maximum size prediction in Wicksell's corpuscle problem for the exponential tail data (Q1424666)

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.