Inference for stereological extremes.

The occurrence of imperfections - inclusions - is unavoidable in the production of clean steels. The strength of such a product is thought to be largely dependent on the size of the largest imperfection it contains, so inference on extreme inclusion size forms an important part of quality control for clean steels. Sampling is generally done by measuring imperfections on planar slices, leading to an extreme value version of a standard stereological problem: what can be said about the process of large inclusions on the basis of the sliced observations? This leads to an extreme value variant of the famous Wicksells corpuscle problem, previously addressed using a combination of standard extreme value models, stereological calculations and a Bayesian hierachical model formulation. However, previous analyses have generally been made under the assumption of spherical inclusions. In this article we consider the sensitivity of results to this assumption, allowing a broader class of inclusion shapes. This is not a trivial matter as the standard stereological calculations are valid only in the spherical case, leading to difficulties in likelihood specification. We focus therefore on likelihood-free Markov chain Monte Carlo methods, to which we add some ideas from simulated tempering procedures, to obtain an approximate inference. The methodology is shown to give good agreement with exact MCMC inference on restriction to spherical inclusions, and is checked more generally via a simulation study. On application to the steel data it is found that results are sensitive to the spherical assumption, but in a somewhat limited way.