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On[alpha]-Symmetric Multivariate Characteristic Functions

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  • Gneiting, Tilmann

Abstract

Ann-dimensional random vector is said to have an[alpha]-symmetric distribution,[alpha]>0, if its characteristic function is of the form[phi]((u1[alpha]+...+un[alpha])1/[alpha]). We study the classes[Phi]n([alpha]) of all admissible functions[phi]: [0, [infinity])-->. It is known that members of[Phi]n(2) and[Phi]n(1) are scale mixtures of certain primitives[Omega]nand[omega]n, respectively, and we show that[omega]nis obtained from[Omega]2n-1byn-1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: If[phi](0)=1,[phi]is continuous, limt-->[infinity] [phi](t)=0, and[phi](2n-2)(t) is convex, then[phi][set membership, variant][Phi]n(1). The paper closes with various criteria for the unimodality of an[alpha]-symmetric distribution.

Suggested Citation

  • Gneiting, Tilmann, 1998. "On[alpha]-Symmetric Multivariate Characteristic Functions," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 131-147, February.
    • Handle: RePEc:eee:jmvana:v:64:y:1998:i:2:p:131-147
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    References listed on IDEAS

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    1. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    2. Kellerer H., 1966. "Ein Beitrag zur Wissenschaftsstatistik," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 179(1), pages 445-451, February.
    3. Berk, Robert & Hwang, Jiunn T., 1989. "Optimality of the least squares estimator," Journal of Multivariate Analysis, Elsevier, vol. 30(2), pages 245-254, August.
    4. Richards, Donald St. P., 1986. "Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform," Journal of Multivariate Analysis, Elsevier, vol. 19(2), pages 280-298, August.
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    Cited by:

    1. Rustam Ibragimov, 2005. "Portfolio Diversification and Value At Risk Under Thick-Tailedness," Yale School of Management Working Papers amz2386, Yale School of Management, revised 01 Aug 2005.

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