A unified treatment of characteristic functions of symmetric multivariate and related distributions

The purpose of the present paper is to give unified expressions to the characteristic functions of all elliptical and related distributions. Those distributions including the multivariate elliptical symmetric distributions and some asymmetric distributions such as skew-elliptical distributions and their locationscale mixtures. In particular, we get simple closed form of characteristic functions for important cases such as the multivariate Student-t, Cauchy, logistic, Laplace, symmetric stable. The expressions of characteristic functions involve Bessel type functions or generalized hypergeometric series.

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

[2]  O. Arslan.  Variance-Mean Mixture of Kotz-Type Distributions , 2009 .

[3]  C. Yin,et al.  A New Class of Multivariate Elliptically Contoured Distributions with Inconsistency Property , 2020, Methodology and Computing in Applied Probability.

[4]  Marc G. Genton,et al.  Characteristic functions of scale mixtures of multivariate skew-normal distributions , 2011, J. Multivar. Anal..

[5]  M. Genton,et al.  Generalized skew-elliptical distributions and their quadratic forms , 2005 .

[6]  O. Arslan.  Family of multivariate generalized t distributions , 2004 .

[7]  Narayanaswamy Balakrishnan,et al.  An Identity for Expectations and Characteristic Function of Matrix Variate Skew-normalDistribution with Applications to Associated Stochastic Orderings , 2021 .

[8]  Chuancun Yin,et al.  A new class of symmetric distributions including the elliptically symmetric logistic , 2018, Communications in Statistics - Theory and Methods.

[9]  Narayanaswamy Balakrishnan,et al.  Expressions for joint moments of elliptical distributions , 2021, J. Comput. Appl. Math..

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

[11]  A. Joarder.  On the characteristic function of the multivariate Pearson type II distribution , 1997 .

[12]  W. N. Bailey.  Confluent Hypergeometric Functions , 1960, Nature.

[13]  Generalized skew-elliptical distributions are closed under affine transformations , 2018 .

[14]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[15]  N. Balakrishnan,et al.  Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties , 2014, J. Multivar. Anal..

[16]  Saralees Nadarajah,et al.  The Kotz-type distribution with applications , 2003 .

[17]  M. Genton,et al.  On fundamental skew distributions , 2005 .

[18]  Chuancun Yin.  Stochastic orderings of multivariate elliptical distributions , 2021, Journal of Applied Probability.

[19]  Esmat Jamshidi Eini,et al.  The tail mean–variance optimal portfolio selection under generalized skew-elliptical distribution , 2021, Insurance: Mathematics and Economics.

[20]  Christine M. Anderson-Cook,et al.  Book review: quantitative risk management: concepts, techniques and tools, revised edition, by A.F. McNeil, R. Frey and P. Embrechts. Princeton University Press, 2015, ISBN 978-0-691-16627-8, xix + 700 pp. , 2017, Extremes.

[21]  R. Arellano-Valle,et al.  On the Unification of Families of Skew‐normal Distributions , 2006 .

[22]  A. Azzalini,et al.  Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution , 2003, 0911.2342.

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

[24]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

[25]  Mehdi Amiri,et al.  Comparison of the multivariate skew-normal random vectors based on the integral stochastic ordering , 2020, Communications in Statistics - Theory and Methods.

[26]  Arjun K. Gupta,et al.  A multivariate skew normal distribution , 2004 .

[27]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[28]  D. Song,et al.  A Note on the Characteristic Function of Multivariate t Distribution , 2014 .

[29]  Raluca Vernic.  Multivariate skew-normal distributions with applications in insurance , 2006 .

[30]  E. Seneta,et al.  A characterisation of scale mixtures of the uniform distribution , 2008 .