THE MAXIMUM SIZE OF THE PLANAR SECTIONS OF RANDOM SPHERES AND ITS APPLICATION TO METALLURGY

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

1Kobe University of Mercantile Marine, 5-1-1, Fukae-Minami, Higashi-Nada-ku, Kobe 658, Japan 2Department 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.