Electron Backscatter Diffraction (EBSD) is a technique used in materials science to study the microtexture of metals, producing data which measure orientations of crystals in a specimen. We examine the precision of such data based on a useful class of distributions on orientations in 3 dimensions (as represented by 3 × 3 orthogonal matrices with positive determinant). While such modeling has received attention in the statistical literature, the approach taken has typically been from general "special manifold" considerations and the resulting methodology may not be easily accessible to non-specialists. We take a more direct modeling approach, beginning from a simple intuitively appealing mechanism for generating random orientations specifically in 3-space. The resulting class of distributions has many desirable properties, including directly interpretable parameters and relatively simple theory.We investigate the basic properties of the entire class and one-sample quasi-likelihood-based 1 inference for one member of the model class, producing new statistical methodology that proves practically useful in the analysis of EBSD data.

The Supplemental Materials provided here consist of two parts. Proofs of the distributional results, regarding the coverage properties of out-of-bag prediction intervals (Section 3 of the main manuscript), are detailed in Section S.1. In Section S.2 and Section S.3, we then provide some additional figures to further support the numerical studies given in Sections 5-6 of the main manuscript.