Stereological estimation for particle processes and random tessellations

Abstract

In this thesis we develop statistical methodology for stereological estimation problems which in particular appear in the field of materials science. We study mathematical models which may describe materials microstructures, and we develop statistical methods for estimating the parameter(s) of these models in a stereological setting. That is, typically we consider a 3D model, and instead of directly observing data generated by this model we only observe a 2D planar section. In many cases we even consider (𝑑 −1)-dimensional sections of a 𝑑-dimensional model (𝑑 = 2, 3, . . . ), because it turns out this is often not more complicated than the 𝑑 = 3 case. In order to formally define and study these models we mainly rely on concepts from stochastic geometry and point process theory. This thesis is divided into two parts. In part I we consider a model of randomly positioned and -oriented particles of the same shape, but varying in size, and in part II we consider a space-filling model by studying a random tessellation (mosaic). In these models, the particles of a particle process, or the cells of a random tessellation, may represent the so-called grains of a materials microstructure....