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THE MAXIMUM SIZE OF THE PLANAR SECTIONS OF

RANDOM SPHERES AND ITS APPLICATION TO METALLURGY

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RINYA TAKAHASHI^{ 1} AND MASAAKI SIBUYA^{ 2}

^{1} *Kobe University of Mercantile Marine, 5-1-1,
Fukae-Minami,*

Higashi-Nada-ku, Kobe 658, Japan

^{2} *Department of Mathematics, Keio University, 3-14-1,
Hiyoshi,*

Kohoku-ku, Yokohama 223, Japan
(Received June 17, 1994; revised April 12, 1995)

**Abstract.**
A theorem of this paper proves that if the size
distribution of random spheres is generalized gamma, its Wicksell
transform and other related distributions belong to the domain of
attraction of the Gumbel distribution. The theorem also shows the
attraction coefficients of the distributions. The fatigue strength
of high-strength steel is closely related to the maximum size of
nonmetallic inclusions in the region of maximum stress of the steel.
Murakami and others developed a method, making use of the
Gumbel QQ-plot, for predicting the maximum size from the size
distribution of inclusion circles in microscopic view-fields. Based
on the Gumbel approximation of the maximum of Wicksell transforms, a
modified and extended version of Murakami's method is justified, and
its performance is evaluated by simulation.

*Key words and phrases*:
Extreme value theory, generalized
gamma distribution, Gumbel distribution, metal fatigue, stereology,
Wicksell's corpuscle problem.

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