PREDICTION OF THE MAXIMUM SIZE IN WICKSELL’S CORPUSCLE PROBLEM, II

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

1Kobe University of Mercantile Marine, 5-1-1, Fukae-Minami, Higashi-Nada-ku, Kobe 658-0022, Japan 2Takachiho University, 2-19-1 Ohmiya, Suginami-ku, Tokyo 168-8508, Japan

(Received May 28, 1998; revised March 9, 2000)

Abstract. This is a continuing paper of the authors (1998, Ann. Inst. Statist. Math., 50, 361–377). 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 three-parameter generalized gamma distribution. Prediction methods based on the moment estimation are proposed and their performances are evaluated by simulation. For a practically probable case, one of these prediction methods is as good as a method previously proposed by the authors where the two shape parameters are assumed to be known.

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