A nonparametric permutation approach to statistical shape analysis

The statistical community has shown an increased interest in shape analysis in the last decade, in particular with reference to the development of robust inferential statistical methods. In this Ph.D. thesis we present an extension of NonParametric Combination (NPC) methodology (Pesarin, 2001) to shape analysis. At first we review inferential methods known in the shape analysis literature, highlighting some drawbacks of using Hotelling's T^2 test statistic. Then, focussing on the two independent sample case, through an exhaustive comparative simulation study, we evaluate the behaviour of traditional tests along with nonparametric permutation tests using also Multiple Aspect (MA) procedures and domain combinations. The case of heterogeneous and dependent variation at each landmark is also investigated, along with the effects of superimposition on the power of NPC tests. Permutation tests have been evaluated also in the particular case in which the number of variables is larger than the cardinality of permutation sample space. We have performed a simulation study to evaluate the power of multivariate NPC tests, showing that the power for the proposed tests increases when increasing the number of the processed variables provided that the noncentrality parameter increases, even when the number of covariates is larger than the permutation sample space. These preliminary results allowed us to extend the notion of finite-sample consistency for permutation tests combination-based to the shape analysis field. Sufficient conditions are given in order that the rejection rate converges to one, for fixed sample sizes at any attainable alpha-value, when the number of variables diverges, provided that the noncentrality induced by test statistics also diverges. On the basis of these findings, we emphasize that the proposed tests provide efficient solutions to multivariate small sample problems, like those encountered in the shape analysis field. Along with simulation studies, we present two applications to real data sets concerning Mediterranean monk seal skulls and aortic valve morphology.