Isotonic regression for metallic microstructure data: estimation and testing under order restrictions

Investigating the main determinants of the mechanical performance of metals is not a simple task. Already known physically inspired qualitative relations between 2D microstructure characteristics and 3D mechanical properties can act as the starting point of the investigation. Isotonic regression allows to take into account ordering relations and leads to more efficient and accurate results when the underlying assumptions actually hold. The main goal in this paper is to test order relations in a model inspired by a materials science application. The statistical estimation procedure is described considering three different scenarios according to the knowledge of the variances: known variance ratio, completely unknown variances, and variances under order restrictions. New likelihood ratio tests are developed in the last two cases. Both parametric and non-parametric bootstrap approaches are developed for finding the distribution of the test statistics under the null hypothesis. Finally an application on the relation between geometrically necessary dislocations and number of observed microstructure precipitations is shown.