Consistency of Procrustes Estimators

Lele has shown that the Procrustes estimator of form is inconsistent and raised the question about the consistency of the Procrustes estimator of shape. In this paper the consistency of estimators of form and shape is studied under various assumptions. In particular, it is shown that the Procrustes estimator of shape is consistent under the assumption of an isotropic error distribution and that consistency breaks down if the assumption of isotropy is relaxed. The relevance of these results for practical shape analysis is discussed. As a by-product, some new results are derived for the offset uniform distribution from directional data.