New Results on Truncated Elliptical Distributions

Truncated elliptical distributions occur naturally in theoretical and applied statistics and are essential for the study of other classes of multivariate distributions.Two members of this class are the multivariate truncated normal and multivariate truncated t distributions.We derive statistical properties of the truncated elliptical distributions.Applications of our results establish new properties of the multivariate truncated slash and multivariate truncated power exponential distributions.

Projected Elliptical Distributions

Abstract We introduce a new parametrization of elliptically contoured densities and study the associated family of projected (circular) distributions. In particular we investigate the trigonometric moments and some convolution properties.

THE EXPECTED VALUES OF INVARIANT POLYNOMIALS WITH MATRIX ARGUMENT OF ELLIPTICAL DISTRIBUTIONS

Invariant polynomials with matrix arguments have been defined by the theory of group representation, generalizing the zonal polynomials. They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interesting polynomials has been shown by people working in the field of econometric theory. In this paper. we derive the expected values of C (BR.BU). Ck (BR)C (BU) and Ck (B-1U), where Bd＝X′X and Xnxp is distributed according to an elliptical matrix distribution. We also give their applications in multivariate distribution theory including the related development in econometrics.

ESTIMATION OF SCATTER MATRIX BASED ON i．i．d． SAMPLE FROM ELLIPTICAL DISTRIBUTIONS

In this paper, we consider the estimation of a scatter matrix under entropy loss, quadratic loss, when the samples x(1),…,x(n) are i.i.d. and x(1)￣ECp(μ,Σ,f). With respect to entropy and quadratic losses, we obtain the best estimator of Σ having the form αSx as well as having the form Tx△Tx' , where Sx,Tx and △ are given in the text, and obtain the minimax estimator of Σ and the best equivariant estimator of Σ with respect to the triangular transformations group LT+ (p)(the group consisting of lower triangular matrices with positive diagonal elements). Some related discussion are given as its generalizations.

Impact of Eccentricity on Nonlinear Oscillations of a Point-Like Charge in the Electric Field of a Curvature-Dependent Elliptic Charged Ellipse

Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Although challenging, nonetheless the scope of the study limited to uniform charge distributions of the rings. Here we extend the analysis considering a charged ellipse with a nonuniform, curvature-dependent elliptic charge distribution exerting a force on a point-like charge placed on the vertical symmetry axis. Nonuniform charge distribution and its impact on various practical scenarios are not a common theme addressed in literature. Applying Computer Algebra System (CAS) particularly Mathematica [2], we analyze the issue on hand augmenting the traditional scope of interest. We overcome the CPU expensive symbolic computation following our newly developed numeric/symbolic method [1]. For comprehensive understanding, we simulate the nonlinear oscillations.

Some Comments on Zonal Polynomials and Their Expected Values with respect to Elliptical.Distributions

In this paper, we give alternative proofs of some results in [15] （Li R.,1997） about the expected value of zonal polynomials.

Shape Retrieval Using ECPDH with Dynamic Programming

The matching and retrieval of the 2D shapes are challenging issues in object recognition and computer vision. In this paper, we propose a new object contour descriptor termed ECPDH (Elliptic Contour Points Distribution Histogram), which is based on the distribution of the points on an object contour under the polar coordinates. ECPDH has the essential merits of invariance to scale and translation. Dynamic Programming (DP) algorithm is used to measure the distance between the ECPDHs. The effectiveness of the proposed method is demonstrated using some standard tests on MPEG-7 shape database. The results show the precision and recall of our method over other recent methods in the literature.

Nonlinear Electrostatic “Hesitant” Oscillator

In search of nonlinear oscillations, we envision a 3D elliptic curva-ture-dependent nonuniform charge distribution to creating an electric field along the symmetry axis causing a massive point-like charged particle placed on the symmetry axis to oscillate in a delayed/hesitant nonlinear mode. The charge distribution is a 3D twisted line creating nontrivial electric field causing an unexpected oscillation that is non-orthodox defying the common sense. Calculation of this research flavored investigation is entirely based on utilities accompanied with Computer Algebra Systems (CAS) especially Mathematic [1]. The characteristics of the delayed oscillations in addition to embodying classic graphics displaying the time-dependent kinematic quantities are augmented including various phase diagrams signifying the nonlinear oscillations. The output of our investigation is compared to nonlinear non-delayed oscillations revealing fresh insight. For comprehensive understanding of the hesitant oscillator a simulation program is crafted clarifying visually the scenario on hand.

Heteroscedasticity and/or Autocorrelation Checks in Longitudinal Nonlinear Models with Elliptical and AR(1) Errors

The aim of this paper is to study the tests for variance heterogeneity and/or autocorrelation in nonlinear regression models with elliptical and AR（1） errors. The elliptical class includes several symmetric multivariate distributions such as normal, Student-S, power exponential, among others. Several diagnostic tests using score statistics and their adjustment are constructed. The asymptotic properties, including asymptotic chi-squave and approximate powers under local alternatives of the score statistics, are studied. The properties of test statistics are investigated through Monte Carlo simulations. A data set previously analyzed under normal errors is reanalyzed under elliptical models to illustrate our test methods.

High-dimensional proportionality test of two covariance matrices and its application to gene expression data

With the development of modern science and technology, more and more high-dimensionaldata appear in the application fields. Since the high dimension can potentially increase the com-plexity of the covariance structure, comparing the covariance matrices among populations isstrongly motivated in high-dimensional data analysis. In this article, we consider the proportion-ality test of two high-dimensional covariance matrices, where the data dimension is potentiallymuch larger than the sample sizes, or even larger than the squares of the sample sizes. We devisea novel high-dimensional spatial rank test that has much-improved power than many exist-ing popular tests, especially for the data generated from some heavy-tailed distributions. Theasymptotic normality of the proposed test statistics is established under the family of ellipticallysymmetric distributions, which is a more general distribution family than the normal distribu-tion family, including numerous commonly used heavy-tailed distributions. Extensive numericalexperiments demonstrate the superiority of the proposed test in terms of both empirical sizeand power. Then, a real data analysis demonstrates the practicability of the proposed test forhigh-dimensional gene expression data.