Bootstrap confidence regions for the planar mean shape

We consider bootstrap methods for constructing confidence regions for the mean shape of objects specified by labelled landmarks in two dimensions. Two statistics are considered: a pivotal statistic, T, derived using matrix perturbation arguments; and a Hotelling-type statistic, H, based on partial Procrustes tangent projections of the observations. We give a rigorous proof, under weak conditions, that the null asymptotic distribution of T is χ2. Simulation results show that (i) the confidence region procedure obtained by bootstrapping each statistic is clearly superior to the corresponding 'tabular' procedure; and (ii) the pivotal T bootstrap confidence regions generally have smaller coverage error than the Hotelling bootstrap confidence regions, especially for distributions with low concentration.