Nonparametric confidence regions for the central orientation of random rotations

Three-dimensional orientation data, with observations as 3×3 rotation matrices, have applications in areas such as computer science, kinematics and materials sciences, where it is often of interest to estimate a central orientation parameter S represented by a 3×3 rotation matrix. A well-known estimator of this parameter is the projected arithmetic mean and, based on this statistic, two nonparametric methods for setting confidence regions for S exist. Both of these methods involve large-sample normal theory, with one approach based on a data-transformation of rotations to directions (four-dimensional unit vectors) prior to analysis. However, both of these nonparametric methods may result in poor coverage accuracy in small samples. As a remedy, we consider two bootstrap methods for approximating the sampling distribution of the projected mean statistic and calibrating nonparametric confidence regions for the central orientation parameter S. As with normal approximations, one bootstrap method is based on the rotation data directly while the other bootstrap approach involves a data-transformation of rotations into directions. Both bootstraps are shown to be valid for approximating sampling distributions and calibrating confidence regions based on the projected mean statistic. A simulation study compares the performance of the normal theory and proposed bootstrap confidence regions for S, based on common data-generating models for symmetric orientations. The bootstrap methods are shown to exhibit good coverage accuracies, thus providing an improvement over normal theory approximations especially for small sample sizes. The bootstrap methods are also illustrated with a real data example from materials science.