Bootstrap inference for mean reflection shape and size-and-shape with three-dimensional landmark data

Working within the framework of a multi-dimensional scaling approach to shape analysis, we develop bootstrap methods for inference about mean reflection shape and size-and-shape based on labelled landmark data. The approach is developed in general dimensions though we focus on the three-dimensional case. We consider two pivotal statistics which we use to construct bootstrap confidence regions for the mean reflection shape or size-and-shape, and present simulation results which show that these statistics perform well in a variety of examples. We also suggest regularized versions of the test statistics that are suitable for more challenging cases where sample size is not sufficiently large in relation to the number of landmarks and present numerical results confirming that regularization indeed leads to better performance. An algorithm for producing a graphical representation of the confidence region for the mean reflection shape is presented and applied in an example involving molecular dynamics simulation data.