Elliptically Contoured Distributions

The elliptically contoured distribution (ECD for short) is a class of multivariate distributions that are considered as an extension of the multivariate normal distributions. In this entry, we introduce a historical perspective, basic properties, and subclasses of the ECD. The estimation of parameters of an ECD implies the study of elliptical matrix distributions. A brief review on elliptical matrix distributions is given. Some applications of ECD are discussed.

[1]  R. Muirhead,et al.  Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations , 1980 .

[2]  Steffen L. Lauritzen,et al.  Multivariate Dispersion Models , 2000 .

[3]  Kai-Tai Fang,et al.  Some Applications of Läuter's Technique in Tests for Spherical Symmetry , 2000 .

[4]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[5]  M. S. Bartlett,et al.  The vector representation of a sample , 1934, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Kai-Tai Fang,et al.  Maximum‐likelihood estimates and likelihood‐ratio criteria for multivariate elliptically contoured distributions , 1986 .

[7]  D. Dey,et al.  A General Class of Multivariate Skew-Elliptical Distributions , 2001 .

[8]  Aurel Wintner,et al.  On the Spherical Approach to the Normal Distribution Law , 1940 .

[9]  James Allen Fill,et al.  On Projection Pursuit Measures of Multivariate Location and Dispersion , 1984 .

[10]  S. Kotz,et al.  Characteristic Functions of a Class of Elliptic Distributions , 1994 .

[11]  David E. Tyler.  Robustness and efficiency properties of scatter matrices , 1983 .

[12]  S. Kotz,et al.  The Meta-elliptical Distributions with Given Marginals , 2002 .

[13]  M. C. Jones.  Marginal Replacement in Multivariate Densities, with Application to Skewing Spherically Symmetric Distributions , 2002 .

[14]  A largest characterization of spherical and related distributions , 1991 .

[15]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[16]  James Clerk Maxwell,et al.  V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres , 1860 .

[17]  Runze Li,et al.  Bayesian Statistical Inference on Elliptical Matrix Distributions , 1999 .

[18]  David E. Tyler.  Radial estimates and the test for sphericity , 1982 .

[19]  Tatsuya Kubokawa,et al.  Robust Improvement in Estimation of a Mean Matrix in an Elliptically Contoured Distribution , 2001 .

[20]  M. A. Chmielewski,et al.  Elliptically Symmetric Distributions: A Review and Bibliography , 1981 .

[21]  C. S. Wong,et al.  Moments for Left Elliptically Contoured Random Matrices , 1994 .

[22]  Mark E. Johnson.  Multivariate Statistical Simulation: Johnson/Multivariate , 1987 .

[23]  Ludwig Baringhaus,et al.  Limit Distributions for Mardia's Measure of Multivariate Skewness , 1992 .

[24]  Dietrich von Rosen,et al.  Moments for matrix normal variables , 1988 .

[25]  J. F. Wagner,et al.  Elliptically symmetric distributions , 1968, IEEE Trans. Inf. Theory.

[26]  M. L. Eaton.  A characterization of spherical distributions , 1986 .

[27]  P. Bentler,et al.  A necessary test of goodness of fit for sphericity , 1993 .

[28]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[29]  The Sample Covariance Is Not Efficient for Elliptical Distributions , 2002 .