Estimation of mean form and mean form difference under elliptical laws

TL;DR

This paper generalizes the estimation of mean form and mean form difference from Gaussian to elliptical distributions, revising previous models, deriving exact moments, and applying the methods to real landmark data and handwritten differentiation.

Contribution

It extends the Gaussian perturbation model to elliptical laws, providing exact formulas for moments and improving estimation and hypothesis testing methods.

Findings

Revised inaccuracies in the perturbation model.

Derived exact moments for matrix B.

Applied methods to real landmark data and handwriting analysis.

Abstract

Some ideas studied by Lele (1993), under a Gaussian perturbation model, are generalised in the setting of matrix multivariate elliptical distributions. In particular, several inaccuracies in the published statistical perturbation model are revised. In addition, a number of aspects about identifiability and estimability are also considered. Instead of using the Euclidean distance matrix for proposing consistent estimates, this paper determines exact formulae for the moments of matrix $\mathbf{B} = \mathbf{X}^{c}\left(\mathbf{X}^{c}\right)^{T}$, where $\mathbf{X}^{c}$ is the centered landmarks matrix. Consistent estimation of mean form difference under elliptical laws is also studied. Finally, the main results of the paper and some methodologies for selecting models and hypothesis testing are applied to a real landmark data. comparing correlation shape structure is proposed and applied in handwritten differentiation.