Estimation of mean form and mean form difference under elliptical laws

The matrix variate elliptical generalization of [30] is presented in this work. The published Gaussian case is revised and modified. Then, new aspects of identifiability and consistent estimation of mean form and mean form difference are considered under elliptical laws. For example, instead of using the Euclidean distance matrix for the consistent estimates, exact formulae are derived for the moments of the matrix B = Xc(Xc)T; where Xcis the centered landmark matrix. Finally, a complete application in Biology is provided; it includes estimation, model selection and hypothesis testing. © 2017, Institute of Mathematical Statistics. All rights reserved

The impossibility of a recurrence construction of the invariant polynomials by using the laplace-beltrami operator

In this paper, we solve an open problem proposed by Davis in [3] about the construction of the invariant polynomials by using the Laplace-Beltrami operator. Until now, only a basis for the non-normalized polynomials is known, and the coefficients need to be collected in τ × τ array of integers. The required solution demanded the construction of a new basis, in certain subspace, which can be written in a triangular array of ρ (ρ + 1) 2 non-negative integers (ρ ≤ τ), the method also provides some useful checking rules for the correctness of the tables. However, a counterexample for a recursion method is provided, going against the old conjecture that Davis’ polynomials can be computed recurrently as James’ zonal polynomials (James [7]). Given that the method is defined for the eigenvalues of the implied positive definite matrices, the idea of the paper holds naturally for invariant polynomials under real normed division algebras (real, complex, quaternions and octonions). © 2016 Pushpa Publishing House, Allahabad, India

A family of formulae for pi

A simple circle method for constructing rational points and tangents with straightedge and compass is introduced; taking the limit, this leads to a family of N formulae for π. The whole process is based on the derivation of a Pythagorean triple and the construction of the circle by irregular inscribed polygons with rational area. © 2012 Pushpa Publishing House

Asymptotic normality of the optimal solution in multiresponse surface mathematical programming

An explicit form for the perturbation effect on the matrix of regression coefficients on the optimal solution in multiresponse surface methodology is obtained in this paper. Then, the sensitivity analysis of the optimal solution is studied and the critical point characterisation of the convex program, associated with the optimum of a multiresponse surface, is also analysed. Finally, the asymptotic normality of the optimal solution is derived by the standard methods

Matrix variate Birnbaum–Saunders distribution under elliptical models

This paper derives the elliptical matrix variate version of the well known univariate Birnbaum–Saunders distribution of 1969. A generalisation based on a matrix transformation is proposed, instead of the independent element-to-element elliptical extension of the Gaussian univariate case. Some results on Jacobians were needed to derive the new matrix variate distribution. A number of particular distributions are studied and some basic properties are found. Finally, an example based on real data of two populations is given and the maximum likelihood estimates are obtained for the class of Kotz models. A comparison with the Gaussian kernel is also given by using a modified BIC criterion. © 2020 Elsevier B.V

Matrix Kummer-Pearson VII relation and polynomial pearson VII configuration density

A case of the matrix Kummer relation of Herz (1955) based on the Pearson VII type matrix model is derived in this paper. As a consequence, the polynomial Pearson VII configuration density is obtained and this sets the corresponding exact inference as a solvable aspect in shape theory. An application in postcode recognition, including a numerical comparison between the exact polynomial and the truncated configuration density, is given at the end of the paper

Statistical theory of shape under elliptical models via QR decomposition

The statistical shape theory via QR decomposition and based on Gaussian and isotropic models is extended in this paper to the families of non-isotropic elliptical distributions. The new shape distributions are easily computable and then the inference procedure can be studied with the resulting exact densities. An application in Biology is studied under two Kotz models, the best distribution (non-Gaussian) is selected by using a modified Bayesian information criterion (BIC)*. © 2013 © 2013 Taylor & Francis

Elliptical affine shape distributions for real normed division algebras

The statistical affine shape theory is set in this work in the context of real normed division algebras. The general densities apply for every field: real, complex, quaternion, octonion, and for any noncentral and non-isotropic elliptical distribution; then the separated published works about real and complex shape distributions can be obtained as corollaries by a suitable selection of the field parameter and univariate integrals involving the generator elliptical function. As a particular case, the complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients

Matrix generalised Kummer relation

An extension of the well known matrix Kummer relation of Herz (1955) is proposed in this paper by assuming a general model which admits a Taylor expansion. Under certain conditions of the involved parameters, the addressed generalisation can be used for deriving efficiently computable matrix variate densities based on non Gaussian models, avoiding some open problems of the literature in the context of shape theory and other related fields. © Publishe