Kybernetika 42 no. 1, 77-94, 2006

Extremes of spheroid shape factor based on two dimensional profiles

Daniel Hlubinka


The extremal shape factor of spheroidal particles is studied. Three dimensional particles are considered to be observed via their two dimensional profiles and the problem is to predict the extremal shape factor in a given size class. We proof the stability of the domain of attraction of the spheroid's and its profile shape factor under a tail equivalence condition. We show namely that the Farlie-Gumbel-Morgenstern bivariate distributions gives the tail uniformity. We provide a way how to find normalising constants for the shape factor extremes. The theory is illustrated on examples of distributions belonging to Gumbel and Fréchet domain of attraction. We discuss the ML estimator based on the largest observations and hence the possible statistical applications at the end.