LOCAL STEREOLOGY OF EXTREMES
DOI:
https://doi.org/10.5566/ias.v31.p99-108Keywords:
extremes, local stereology, maximum domain of attraction, shape and size parameters, spheroidsAbstract
Local stereology uses information obtained from central sections passing through a reference point of the particle.The aim of this paper is to investigate the prediction of extremes of shape and size parameters based on the central sections. We consider the particle population formed by spheroids (either prolate or oblate) and assume that the reference point is the centre of the spheroid. A relation between shape and size parameters of the particles and their planar sections is derived and consequently stability properties of the domain of attractions are proved.References
bibitem[Benev{s} and Rataj(2004)]{br04} Benev{s} V and Rataj J (2004). Stochastic Geometry: Selected Topics. Kluwer Academic Publishers, Boston.
bibitem[Bortot etal(2007)]{bcs07} Bortot P, Coles SG and Sisson SA (2007). Inference for stereological extremes. J. Amer. Statist. Assoc. 102:84--92.
bibitem[de Haan(1975)]{dH75} de Haan L (1975). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, 3rd printing. Mathematical Centre Tracts 32, Mathematisch Centrum, Amsterdam.
bibitem[Drees and Reiss(1992)]{dr92} Drees H and Reiss RD (1992).
Tail behaviour in Wicksell's corpuscle problem. In: Probability Theory
and Applications -- Essays to the Memory of J'ozsef Megyor'odi,
Galambos J and K'atai I (eds.), 205--220. Kluwer Academic Publishers,
Dordrecht.
bibitem[Embrechts etal(1997)]{ekm97} Embrechts P, Kl"uppelberg C and Mikosch T (1997). Modelling Extremal Events. Springer, Berlin.
bibitem[Hlubinka(2003a)]{h03a} Hlubinka D (2003). Stereology of extremes; shape factor of spheroids. Extremes 6:5--24.
bibitem[Hlubinka(2003b)]{h03b} Hlubinka D (2003). Stereology of extremes; size of spheroids. Math. Bohem. 128:419--438.
bibitem[Hlubinka(2006a)]{h06a} Hlubinka D (2006). Extremes of spheroid shape factor based on two dimensional profiles. Kybernetika 62:77--94.
bibitem[Hlubinka(2006b)]{h06b} Hlubinka D (2006). Shape factor extremes for prolate spheroids. Kybernetika 62:557--568.
bibitem[Hlubinka(2006c)]{h06c} Hlubinka D (2006). Size and shape factor extremes of spheroids. Image Anal. Stereol. 25:145--154.
bibitem[Hlubinka and Kotz(2010)]{hk10} Hlubinka D and Kotz S (2010). The generalized FGM distribution and its application to stereology of extremes. Appl. Math. 55:495--512.
bibitem[K"otzer and Molchanov(2006)]{km06} K"otzer S and Molchanov I (2006). On the domain of attraction for the lower tail in Wicksell's corpuscle problem. In: Proceedings S$^4$G, Lechnerov'a R, Saxl I and Benev s V (eds.), 91--96. Union of Czech Mathematicians and Physicists, Prague.
bibitem[Molchanov(2005)]{m05} Molchanov I (2005). Theory of Random Sets. Springer, London.
bibitem[Pawlas (2011)]{p11} Pawlas Z (2011). Local stereology of extremes. In: Proceedings of the 13th International Congress for Stereology, Guoquan Liu et al. (eds.), Chinese Society for Stereology, Beijing.
bibitem[Pawlas etal(2009)]{pnvj09} Pawlas Z, Nyengaard JR and Vedel Jensen EB (2009). Particle sizes from sectional data. Biometrics 65:216--224.
bibitem[Takahashi and Sibuya(1996)]{ts96} Takahashi R and Sibuya M (1996). The maximum size of the planar sections of random spheres and its application to metallurgy. Ann. Inst. Statist. Math. 48:127--144.
bibitem[Takahashi and Sibuya(1998)]{ts98} Takahashi R and Sibuya M (1998). Prediction of the maximum size in Wicksell's corpuscle problem. Ann. Inst. Statist. Math. 50:361--377.
bibitem[Takahashi and Sibuya(2001)]{ts01} Takahashi R and Sibuya M (2001). Prediction of the maximum size in Wicksell's corpuscle problem, II. Ann. Inst. Statist. Math. 53:647--660.
bibitem[Takahashi and Sibuya(2002)]{ts02} Takahashi R and Sibuya M (2002). Metal fatigue, Wicksell transform and extreme values. Appl. Stochastic Models Bus. Ind. 18:301--312.
bibitem[Vedel Jensen(1998)]{ebvj98} Vedel Jensen EB (1998). Local
Stereology. World Scientific, Singapore.
bibitem[Weissman(1978)]{wei78} Weissman I (1978). Estimation of parameters and larger quantiles based on the $k$ largest observations. J. Amer. Statist. Assoc. 73:812--815.
bibitem[Wicksell(1925)]{w25} Wicksell SD (1925). The corpuscle problem. A mathematical study of a~biometric problem. Biometrika 17:84--99.
