PREDICTION OF THE MAXIMUM SIZE IN WICKSELL’S CORPUSCLE PROBLEM

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

1Kobe University of Mercantile Marine, 5-1-1, Fukae-Minami, Higashi-Nada-ku, Kobe 658-0022, Japan 2Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

(Received November 29, 1995; revised May 6, 1997)

Abstract. In the Wicksell corpuscle problem, the maximum size of random spheres in a volume part is to be predicted from the sectional circular distribution of spheres cut by a plane. The size of the spheres is assumed to follow the generalized gamma distribution. Some prediction methods according to measurement methods on the sectional plane are proposed, and their performances are evaluated by simulation. The prediction method based on the r largest sizes and the total number of the sectional circles is recommended, because of its satisfactory performance.

Key words and phrases: Extreme value theory, generalized gamma distribution, Gumbel distribution, metal fatigue, stereology.