arXiv subjects
Publications and source records attributed to José A. Díaz-García.
A generalisation of the Charnes-Cooper chance-constrained approach is proposed in the setting of the family of elliptically contoured distributions. The new relaxed stochastic linear programming is notably invariant under the entire class of probability distributions.
This work studies the distribution of the nonsymmetric matrix $\mathbf{E}^{-1}\mathbf{H}$. This random product is of fundamental interest under the general multivariate linear hypothesis setting. Specifically when $\mathbf{H}$ and $\mathbf{E}$ are seen as the sums of squares and the sums of products due to the hypothesis and due to the error, respectively.
In this paper we provide a matrix extension of the scalar binomial series under elliptical contoured models and real normed division algebras. The classical hypergeometric series ${}_{1}F_{0}^β(a;\mathbf{Z})={}_{1}^{k}P_{0}^{β,1}(1:a;\mathbf{Z})=|\mathbf{I}-\mathbf{Z}|^{-a}$ of Jack polynomials are now seen as an invariant generalized determinant with a series representation indexed by any elliptical generator function. In particular, a corollary emerges for a simple derivation of the central matrix variate beta type II distribution under elliptically contoured models in the unified real, complex, quaternions and octonions.
The distribution functions of the matricvariate beta type I and II distributions are studied under real normed division algebras. The unified approach for real, complex, quaternions and octonions, also considers general properties and highlights the potential application of the exact emerging upper probabilities $P(\mathbf{B} > \mathbfΩ)$ and $P(\mathbf{F} > \mathbf{\nabla})$. In this setting, the matrix probabilities arise naturally as univariate extensions into the so termed matrix variate $p$-values. Then, a new criterion for the general multivariate linear hypothesis test can be proposed under a simple heuristic interpretation. The new technique can be applied in a number of classical statistical tests. In particular, the multivariate analysis of variance (MANOVA) is illustrated in two well known scenarios, and the performance of our exact method is compared with the existing approximated criteria.
Recently the termed \emph{multimatrix variate distributions} were proposed in \citet{dgcl:24a} as an alternative for univariate and vector variate copulas. The distributions are based on sample probabilistic dependent elliptically countered models and most of them are also invariant under this family of laws. Despite a large of results on matrix variate distributions since the last 70 years, the spherical multimatrix distributions and the associated probabilities on hyper cones can be computable. The multiple probabilities are set in terms of recurrent integrations allowing several matrix computation a feasible task. An application of the emerging probabilities is placed into a dynamic molecular docking in the SARS-CoV-2 main protease. Finally, integration over multimatrix Wishart distribution provides a simplification of a complex kernel integral in elliptical models under real normed division algebras and the solution was applied in elliptical affine shape theory.
This paper proposes famillies of multimatricvariate and multimatrix variate distributions based on elliptically contoured laws in the context of real normed division algebras. The work allows to answer the following inference problems about random matrix variate distributions: 1) Modeling of two or more probabilistically dependent random variables in all possible combinations whether univariate, vector and matrix simultaneously. 2) Expected marginal distributions under independence and joint estimation of models under likelihood functions of dependent samples. 3) Definition of a likelihood function for dependent samples in the mentioned random dimensions and under real normed division algebras. The corresponding real distributions are alternative approaches to the existing univariate and vector variate copulas, with the additional advantages previously listed. An application for quaternionic algebra is illustrated by a computable dependent sample joint distribution for landmark data emerged from shape theory.
A new family of distributions indexed by the class of matrix variate contoured elliptically distribution is proposed as an extension of some bimatrix variate distributions. The termed \emph{multimatrix variate distributions} open new perspectives for the classical distribution theory, usually based on probabilistic independent models and preferred untested fitting laws. Most of the multimatrix models here derived are invariant under the spherical family, a fact that solves the testing and prior knowledge of the underlying distributions and elucidates the statistical methodology in contrasts with some weakness of current studies as copulas. The paper also includes a number of diverse special cases, properties and generalisations. The new joint distributions allows several unthinkable combinations for copulas, such as scalars, vectors and matrices, all of them adjustable to the required models of the experts. The proposed joint distributions are also easily computable, then several applications are plausible. In particular, an exhaustive example in molecular docking on SARS-CoV-2 presents the results on matrix dependent samples.
This work sets the matrix variate Birnbaum-Saunders theory in the context of singular distributions and elliptical models. The so termed singular matrix variate generalised Birnbaum-Saunders distribution is obtained with respect the Hausdorff measure. Several basic properties and particular cases of this distribution are also derived.
Some general problems of Jacobian computations in non-full rank matrices are discussed in this work. In particular, the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is revisited. Then the Jacobian in the full rank case is derived under the simple and old theory of the exterior product.
A new family of matrix variate distributions indexed by elliptical models are proposed in this work. The so called \emph{multimatricvariate distributions} emerge as a generalization of the bimatrix variate distributions based on matrix variate gamma distributions and independence. Some properties and special cases of the multimatricvariate distributions are also derived. Two new interesting Jacobians in the area are also provided. Finally, an application for time dependent data of DNA molecules is studied.
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.
In this paper a new approach is derived in the context of shape theory. The implemented methodology is motivated in an open problem proposed in \citet{GM93} about the construction of certain shape density involving Euler hypergeometric functions of matrix arguments. The associated distribution is obtained by establishing a connection between the required shape invariants and a known result on canonical correlations available since 1963; as usual in statistical shape theory and the addressed result, the densities are expressed in terms of infinite series of zonal polynomials which involves considerable difficulties in inference. Then the work proceeds to solve analytically the problem of computation by using the Eulerian matrix relation of two matrix argument for deriving the corresponding polynomial distribution in certain parametric space which allows to perform exact inference based on exact distributions characterized for polynomials of very low degree. A methodology for comparing correlation shape structure is proposed and applied in handwritten differentiation.
In this work is obtained an explicit form for the perturbation effect on the matrix of regression coefficients on the optimal solution in multiresponse surface methodology. 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 standard methods.
The non isotropic noncentral elliptical shape distributions via pseudo-Wishart distribution are founded. This way, the classical shape theory is extended to non isotropic case and the normality assumption is replaced by assuming a elliptical distribution. In several cases, the new shape distributions are easily computable and then the inference procedure can be studied under exact densities. An application in Biology is studied under the classical gaussian approach and two non gaussian models.