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Quadratic forms in spherical random variables: Generalized noncentral x2 distribution

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  • T. Cacoullos
  • M. Koutras

Abstract

Let X denote a random vector with a spherically symmetric distribution. The density of U = X'X, called a “generalized chi‐square,” is derived for the noncentral case, when μ = E(X) ≠ 0. Explicit series representations are found in certain special cases including the “generalized spherical gamma,” the “generalized” Laplace and the Pearson type VII distributions. A simple geometrical representation of U is shown to be useful in generating random U variates. Expressions for moments and characteristic functions are also given. These densities occur in offset hitting probabilities.

Suggested Citation

  • T. Cacoullos & M. Koutras, 1984. "Quadratic forms in spherical random variables: Generalized noncentral x2 distribution," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 31(3), pages 447-461, September.
    • Handle: RePEc:wly:navlog:v:31:y:1984:i:3:p:447-461
    DOI: 10.1002/nav.3800310310
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    Cited by:

    1. Marchand, Éric & Strawderman, William E., 2020. "On the non-stochastic ordering of some quadratic forms," Statistics & Probability Letters, Elsevier, vol. 163(C).

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